blO. 9
CMr
CO]*
3
Cajorl History of olemontnry mathematics
kansas city
public library
KANSAS
CITY. M<
BLIC LIBRARY
A HISTORY OF ELEMENTARY MATHEMATICS
BOOKS BY FLORIAN CAJORI HISTORY OF MATHEMATICS Revised and Enlarged Edition
HISTORY OF ELEMENTARY MATHEMATICS Revised and Enlarged Edition
HISTORY OF PHYSICS INTRODUCTION TO THE MODERN
THEORY OF EQUATIONS
A HIS10B2
ELEMENTAKY MATHEMATICS WITH
HINTS ON METHODS OF TEACHING
BT
FLORIAN CAJORI,
PH.D.
PROFESSOR OF THE HISTORY OF MATHEMATICS IN THE UNIVERSITY OF CALIFORNIA
REVISED AND ENLARGED EDITION
THE MACMILLAN COMPANY LONDON
:
MACMILLAN & 1930
CO.,
COPYBIGHT, 1896 AND 1917,
BY THE MACMILLAN COMPANY. no part of this book may be reproduced in any form without permission in writing from
All rights reserved
the publisher.
COPYRIGHT, 1924,
BY PLORIAN CAJORI. Setup and electrotyped. Published September, 1896. Reprinted Feb August, 1897; March, 1905; October, 1907; August, 1910; ruary, 1914
Revised and enlarged edition, February, 1917.
*
PMINTED IN THE UNITED STATES
OiT
AMERICA
PREFACE TO THE FIRST EDITION
"THE
education of the child must accord both in
and arrangement with the education ered historically
or,
;
of
mankind
mode
as consid
in other words, the genesis of knowledge
in the individual must follow the same course as the genesis of knowledge in the race.
To M. Comte we
owes the enunciation of this doctrine
believe society
a doctrine which we
accept without committing ourselves to his theory of 1 the genesis of knowledge, either in its causes or its order." If this principle, held also by Pestalozzi and Proebel, be
may
correct,
then
would seem as
it
history of a science must be an
that science.
Be
if
the knowledge of
effectual aid in
the
teaching
this doctrine true or false, certainly the
experience of many instructors establishes the importance With the hope of of mathematical history in teaching. 2
being of some assistance to niy fellowteachers, I have pre pared this book and have interlined my narrative with occasional remarks
No 1
and suggestions on methods of teaching. reader will draw many useful
doubt, the thoughtful
HJBRBEHT SFBNCEK, Education Intellectual, Moral, and Physical*' York, 1894, p. 122. See also B. H. QUICK, Educational Beformers, :
New
1879, p. 191. 2 See G. HEFFEL, "The Use of History in Teaching Mathematics," Nature, Vol. 48, 18S08, J
1175497
M 21
!946
lessons from the .study of matliematical history which, are f
not directly
(
"pctiitfed
out in the text.
In the preparation of this history, I have made extensive use of the works of Cantor, Hankel, linger, cock,
on
Gow,
and
Allman, Loria,
the history of mathematics.
consulted,
gives
of
other
De Morgan, Pea
prominent writers
Original sources have been
whenever opportunity has presented
me much
pleasure to
itself.
It
acknowledge the assistance ren
dered by the United States Bureau of Education, in for warding for examination a number of old textbooks which otherwise would have been inaccessible to me. also be said that a large
number
&
Co., 1895.
should
of passages in this book
are taken, with only slight alteration, from
Mathematics, Macmillan
It
.
.
my
History of
.
ELOBIAN CAJORI. COLORADO COLLEGE, COLORADO SPRINGS, July, 1896.
PREFACE TO THE SECOND EDITION IN the endeavor
to bring this history
down
to date,
nume*
ous alterations and additions have been made.
FLORIAN CAJOBI. COLORADO COLLEGE, December, 1916.
CONTENTS PA&S
ANTIQUITY
1
NUMBERSYSTEMS AND NUMERALS
1
ARITHMETIC AND ALGEBRA
19
,19
Egypt
26
Greece
.37
Some GEOMETRY AND TRIGONOMETRY
43
Egypt and Babylonia
43
Greece
46
.
Home
89
.93
MIDDLE AGES ARITHMETIC AND ALGEBRA
93
Hindus
93 103
Arabs
Europe during
the Middle
Introduction of
Roman
Ages
Ill
Arithmetic
Ill
Translation of Arabic Manuscripts
The
First
.....
GEOMETRY AND TRIGONOMETRY
122
Hindus Arabs
118
119
Awakening
122
12&
,
Europe during
the Middle
Introduction of
Ages
Roman Geometry
.....
131 131
Translation of Arabic Manuscripts
132
The
134
First
Awakening ttt
CONTENTS
Till
MODERN TIMES ARITHMETIC Its
....... ... ..... ...... ...... ..... ...... ..... ........... ......... ....... ....... .... ... ...... ......
Development as a Science and Art
English Weights and Measures
Rise of the Commercial School in England
.
.
139 139 157
.179
.
Causes which Checked the Growth of Demonstrative Arith metic in England
.
.
204
Reforms in Arithmetical Teaching
211
Arithmetic in the United States
215
" Pleasant and Diverting Questions" AIXJEBRA
219
224
The Renaissance
224
The Last Three Centimes
234
GEOMETBY
AJH> TittIGK)NOMETBT
Editions of Euclid.
Early Researches.
The Beginning of Modem Synthetic Geometry Modern Elementary Geometry
Modern Synthetic Geometry Modern Geometry of the Triangle and NonEuclidean Geometry
.....
Tertbooks on Elementary Geometry
RBCEJST MOVEMENTS IN TEACHING
The Perry Movement
Circle
International Commission .
American Associations
.
.
.
.
Attacks upon the Study of Mathematics the
Mind
......
.
m *
.
257
.
250
.
266
.... .
252
256
.
........ .
245
,
*
.
245
.
.
275 29
2^1
A HISTORY OF MATHEMATICS
NUMBERSYSTEMS NEARLY
all
AJSTD
NUMERALS
nuinbersystems ; both, ancient and modern, are The reason for this it is 5, 10, or 20.
based on the scale of not
When
difficult to see.
a child learns to count, he makes
use of has fingers and perhaps of his toes. In the same way the savages of prehistoric times unquestionably counted on their fingers
and
in
some cases
also
on their
toes.
Such
is
indeed the practice of the African, the Eskimo, and the South This recourse to the fingers has Sea Islander of today. 1 often resulted in the development of a more or less extended pantomime numbersystem, in which the fingers were used as in a deaf and dumb alphabet. 1 Evidence of the prevalence of finger
symbolisms
is
found among the ancient Egyptians,
Babylonians, Greeks, and Eomans,
peans of the middle ages use finger symbolisms. 1
:
even
now
as also
nearly
among the Euro
all
Eastern nations
The Chinese express on the
left
hand
L. L. COSAKT, "Primitive NumberSystems," in Smithsonian
port, 1892, p. 584.
B
1
JKe'
A HISTOEY OF MATHEMATICS
2 "
all numbers less than 100,000 hand touches each joint of the
the external side, then
;
the
thumb
little finger,
nail of the right
passing
first
up
down the middle, and afterwards up
to express the nine digits the it, in order on the second the the same in denoted tens are finger way, hundreds on the third the thousands on the fourth and ten
the other side of
;
;
;
;
thousands on the thumb.
It
would be merely necessary
to
proceed to the right hand in order to be able to extend this l So common is the use of this fingersystem of numeration."
symbolism that traders are said to communicate to one another the price at which they are willing to buy or sell by touching hands, the act being concealed by their cloaks from observa tion of bystanders.
Had
the number of fingers and toes been different in man,
then the prevalent numbersystems of the world would have been different also. We are safe in saying that had one more finger sprouted from each human hand, making twelve fingers in
all,
then the numerical scale adopted by civilized nations
would not be the decimal, but the duodecimal. symbols would be necessary to represent 10 and' tively.
As
far as arithmetic is concerned,
it is
Two more 11, respec
certainly to be
Except for the two more signs or numerals and of being
regretted that a sixth finger did not appear. necessity of using
obliged to learn the multiplication table as far as 12 x 12, the duodecimal system is decidedly superior to the decimal. The
number twelve has for its exact divisors 2, 3, 4, only 2 and o. In ordinary business affairs, the are used extensively,
and
6,
while ten has
fractions
, J,
^
very convenient to have a base which is an exact multiple of 2, 3, and 4. Among the most zealous advocates of the duodecimal scale was Charles XII. it is
GEOBGE PEACOCK, article "Arithmetic," in Encyclop&dia Metropolitana (The Encyclopaedia of Pure Mathematics), p. 394. Hereafter we shall cite this very valuable article as PEACOCK. 1
NUMBBEBYSTBMS AND NUMERALS
3
of Sweden, who, at the time of his death, was contemplating the change for his dominions from the decimal to the duo decimal. l But it is not likely that the change will ever be brought about. So deeply rooted is the decimal system that
when
he storm of the French Revolution swept out of exist ence other old institutions, the decimal system, not only remained unshaken, but was more firmly established than ever.
The advantages
until arithmetic
was
"The
impossible.
of twelve as a base
were not recognized
make a change uncommon one of high
so far developed as to
case
is
the not
civilization bearing evident traces of the rudeness of its origin
in ancient barbaric life."
2
Of the notations based on human anatomy, the quinary and vigesimal systems are frequent among the lower races, while the higher nations have usually avoided the one as too scanty and the other as too cumbrous, preferring the intermediate 3
Peoples have not always consistently adhered to any one scale. In the quinary system, 5, 25, 125, 625, etc., should be the values of the successive higher units >
decimal
system.
but a quinary system thus carried out was never in actual use whenever it was extended to higher numbers it invariably ran eftner into the decimal or into the vigesimal system. "The :
home par
excellence of the quinary, or rather of the quinaryvigesimal scale, is America. It is practically universal among the Eskimo tribes of the Arctic regions. It prevailed among
a considerable portion of the North American Indian tribes, and was almost universal with the native races of Central and 1
CONANT,
op.
eft., p.
589.
E. B. TYLOE, Primitive Culture, New York, 1889, Vol. I., p. 272. In the base 8 or 16, some respects a scale having for its base a power of 2 2
is superior to the duodecimal, but it has the disadvantage for instance, of not being divisible by 3. See W. W. JOHNSON, " Octonary
tion," Bull.
TYLOE,
N. Y. Math. ojp.
Soc., 1891, Yol. L, pp. 16.
cfc, Vol. L, p. 262.
A HISTOBY OF MATHEMATICS
4
South. America. 7 '
l
This system was used also by many of the tribes. Traces of it are found in
North Siberian and African
the languages of peoples who example, in Homeric Greek. traces of it; viz.,
...
now
use the decimal scale
The Eoman
for
;
notation reveals
X, XI, ... XV, etc. It is curious that the quinary should so frequently merge into the vigesimal scale that savages should have passed from the number of fingers on one hand as an upper unit or a stop I, II,
V, VI,
;
pingplace, to the total
unit or restingpoint.
number of fingers and The vigesimal system
toes as an upper is less
common
than the quinary, but, like it, is never found entirely pure. In this the first four units are 20, 400, 8000, 160,000, and special
words for these numbers are actually found among the Mayas of Yucatan. The transition from quinary to vigesimal is shown in the Aztec system, which may be represented thus, 1,2,3, 20 + 1,

5
+ 1,
...10,10 20+10, 20 10
4, 5,
+
occur here for the numbers
+ 1,
...
+ 1,
...
1045,1045 40, etc.
1, 2, 3, 4, 5, 10,
2
+ 1,
...
20,
Special words
20, 40, etc.
The
vigesimal system flourished in America, but was rare in the Old World. Celtic remnants of one occur in the French words
x 20 or x 20 or 300).
quatrevmgts (4 vingts (15
80), stowingts (6 ]>Tote also
x 20
or 120), quinze
the English word score in
such expressions as threescore years and ten. Of the three systems based on human anatomy, the decimal is the most prevalent, so prevalent, in fact, that accord to ancient tradition it was used by all the races of the ing
system world.
It is only within the last
few centuries that the other
1 For further information see also POTT, CONANT, op. c#., p. 592. Die quinare und mgesimale Zahlmethode bei Volkern aller Welttheile, Halle, 1847 POTT, Die jSprachverschiedenhett in Europa an den Zahl* ;
w'ortern nachgewiesen, sowie die quinare Halle, 1868. 2
TTLOB,
op. cit., Vol.
I.,
p. 262.
und mgesimale Zahlmethode,
NUMBERSYSTEMS AST) NUMERALS
5
in use among previously unknown The decimal scale was used in North America by the greater number of Indian tribes, but in South America it
two systems have been found tribes.
was
1
rare.
In the construction of the decimal system, 10 was suggested
by the number of fingers as the first stoppingplace in count ing, and as the first higher unit. Any number between 10 and 100 was pronounced according to the plan &(10) + a(l), a and b being integers less than 10. But the number 110 might be expressed
x
11
Why
10.
The
two ways, (1) as 10 x 10 f 10, (2) as method would not seem unnatural.
in
latter
and say eleventy, instead of between 10 x 10 f 10 and 11 x 10 hinges the systematic construction of the number 2 Good luck led all nations which developed the system. not imitate
hundred and ten
f
eighty, ninetyj
But upon
this choice
decimal system to the choice of the former; 3 the unit 10 being here treated in a manner similar to the treatment of the
lower unit 1 in expressing numbers below 100. Any num ber between 100 and 1000 was designated c(10) 2 f 6(10) f a, a, &, c representing integers less than 10. Similarly for numr bers below 10,000, cZ(10) 3
+ c(10) + fc(lO) 2
1
f a(10)
j
and Simi
larly for still higher numbers.
Proceeding to describe the notations of numbers, we begin with the Babylonian. Cuneiform writing, as also the
accompanying notation of numbers, was probably invented 1
CONANT,
2
HERMANN HANKEL, Zur
op. eft., p. 588.
und Mittelalter, Leipzig, work as HANKEL. 8
GescTiichte
1874, p. 11.
der Mathematik in Alterthum
Hereafter
we
shall cite this brilliant
In this connection read also MOEITZ CANTOR, Vorlesungen uber
Geschichte der Mathematik, Vol. I. (Third Edition), Leipzig, 1907, pp. 6 and 7. This history, by the prince of mathematical historians of the
nineteenth century,
CANTOR.
is in
four volumes, and will be cited hereafter as
A HISTORY OF MATHEMATICS
6 tlie
by
A
early Sumerians.
vertical
wedge
Y
stood
for
^
and y>*~ signified 10 and 100, respectively. In case of numbers below 100, the values of the separate sym while
one,
bols were added.
Thus,
<^T
for 23,
^ ^<
The
for 30.
signs of higher value are written on the left of those of lower But in writing the hundreds a smaller symbol was
value.
placed before that for 100 and was multiplied into 100.
<
signified 10
a If
unit,
^^
10 x 1000.
or 1000.
Taking
Thus,
this for a
new
>
was interpreted, not as 20 x 100, but y In this notation no numbers have been found
1 large as a million.
are the
x 100
additive
The
and the
as as
principles applied in this notation multiplicative.
Besides
Babylonians had another, the sexagesimal
this
notation, to
the be
noticed later.
An
insight into Egyptian methods of notation was obtained through the deciphering of the hieroglyphics by Champollion,
Young, and
S
others.
(1000), f A (10,00 ; 000).
The numerals
^
1
(1),
H
(10), (* (100),
O
(100,000), S" (1,000,000), sign for one represents a vertical staff;
(10,000),
The
are
that for 10,000, a pointing finger; that for 100,000, a burbot; that for 1,000,000, a man in astonishment. No certainty has been reached regarding the significance of the other symbols.
These numerals like the other hieroglyphic signs were plainly animals or objects familiar to the Egyptians, to be conveyed. They
pictures
of
which
some way suggested the idea
in
are excellent examples of picturewriting. The principle in volved in the Egyptian notation was the additive throughout.
Thus, (1> iTjlJ 1
zum
For
would be 111.
fuller treatment see MORITZ CAN
NUMBERSYSTEMS AND NUMERALS local
such as we have in the notation
value,
Having missed
9
now
in use.
had no use
this principle, the ancients
for a
symbol to represent zero, and were indeed very far removed from an ideal notation. In this matter even the Greeks and
Romans failed to achieve what known to Europeans before the
a remote nation in Asia, little
present century, accomplished most admirably. But before we speak of the Hindus, we must speak of an ancient Babylonian notation, which, strange
to say, is not based
on the scale 5, 10, or 20, and which, moreover, came very near a full embodiment of the ideal We refer principle found wanting in other ancient systems, to the sexagesimal notation. x
The Babylonians used
this chiefly in the construction 6t
The systematic development
weights and measures.
of the
both for integers and fractions, reveals a high degree of mathematical insight on the part of the early Sumerians. The notation has been found on two Babylonian sexagesimal
One
tablets.
contains a are 1, 4, 1.40
we
scale,
of them, probably dating
list of
9, 16, 25,
= 10
2
from 1600 or 2300
2 The square numbers up to 60 = 49. We have next 1.4 82 36, .
first
1.21
,
= II
2
B.C.,
seven
=9
2 ,
This remains unintelligible, unless assume the scale of sixty, which makes 1.4 = 60 f 4,
1.21
60
,
2.1
4 21, etc.
,
etc.
The second
tablet records the
magnitude
of the illuminated portion of the moon's disc for every day from new to full moon, the whole disc being assumed to con sist of
240 parts.
The illuminated
days are the series
5,
10, 20, 40,
parts during the
1.20(=
80).
again the sexagesimal scale and also some
first five
This reveals
knowledge of
geometrical progressions. Erom here on the series becomes an arithmetical progression, the numbers from the fifth to the fifteenth day being respectively, 1.20, 1.36, 1.52, 2.8, 2.24, 2.40, In this sexagesimal notation we
2.56, 3.12, 3.28, 3.44, 4.
then, the principle of local value.
have,
Thus, in 1.4 (= 64) ; the 1
is
A HISTORY OF MATHEMATICS
10
made
to stand for 60, the unit of the second order,
by virtue of In Babylonia some use was thus made of the principle of position, perhaps 2000 years before the Hindus developed it. This was at a time when Homulus and Remus, yea even Achilles, Menelaus, and Helen, its
position with respect
to the 4.
unknown to history and song. But the full develop of the principle of position calls for a symbol to represent the absence of quantity, or zero. Babylonian records of about were
still
ment
give a symbol for zero which denoted the absence of a but apparently was not used in calculation. It con figure, About 130 A.D. Ptolemy in his Al sisted of two marks ^
200
B.C.
.
magest used sexagesimal fractions, and also the omieron o to represent blanks in the sexagesimal numbers. This o was not
used as a regular
Hence, the Babyionians had the prin
zero.
and
symbol for^era.to indicate the absence of a figure, but did not use this zero in computation. Their sexagesimal fractions were transmitted to India. ciple of local value,
What was
it
also a
that suggested to the Babylonians the number Cantor 1 and others offer the following
sixty as a base? at 360 days.
At
the Babylonians reckoned the year This led to the division of the circumference of
provisional reply
:
first
a circle into 360 degrees, each degree representing the daily part of the supposed yearly revolution of the sun around the earth. Probably they knew that the radius could be applied to the circumference as a chord six times, and that each arc thus cut off contained 60 degrees. Thus the division
into
60
parts
may have
suggested
itself.
When
greater precision was needed, each degree was divided into 60 equal parts, or minutes. In this way the sexagesimal notation may have originated. The division of the day into 24 hours,
and
of the
hour into minutes and seconds on the i
Vol. L, pp. 9193.
scale of 60,
NUMBERSf STEMS AND NUMERALS due to
is
tlie
11
There are also indications of
Babylonians.
a
1 knowledge of sexagesimal fractions, such, as were used later
by the Greeks, Arabs, by scholars of the middle ages and of even recent times. Babylonian science has made
its
impress upon modern
civili
Whenever
a surveyor copies the readings from the graduated circle on his theodolite, whenever the modern man zation.
notes the time of day, he
unmistakably, doing homage the banks of the Euphrates.
is,
unconsciously perhaps, but
to the ancient astronomers on
The full development of our decimal notation belongs to comparatively modern times. Decimal notation had been in use for thousands of years, before it was perceived that its simplicity and usefulness could be enormously increased by the adoption
of the principle of position.
dus of the
or sixth century after Christ
fifth
To the Hin we owe the
and systematic development of the use of the zero the Of all mathematical dis and principle of local value. to the general progress contributed more coveries, no one has full
of intelligence than this.
While the older notations served
merely to record the answer of an arithmetical computation, the Hindu notation (wrongly called the Arabic notation) assists itself.
first
with 'marvellous power in performing the computation To verify this truth, try to multiply 723 by 364, by
expressing the numbers in the
multiply
DCCXXIII by CCCLXIY.
Eoman This
notation; thus, notation offers
or no help; the Eomans were compelled the aid of the abacus in calculations like this. little
to
invoke
Yery little is known concerning the mode of evolution of the Hindu notation. There is evidence for the belief that the Hindu notation of the second century, A.D., did not include 1
CANTOS, Vol. L,
p. 31,
A HISTORY OF MATHEMATICS
12
On the island of the zero nor the principle of local value. without the zero, but the a notation resembling Hindu,
Ceylon has been preserved. It is known that Buddhism and Indian culture were transplanted thither about the third century and there remained stationary.
It is highly probable, then, that
the old imperfect Hindu system. Besides signs for 19, the Ceylon notation has symbols for each of the tens and for 100 and 1000. Thus the number 7685
the notation of Ceylon
is
would have been written with
six symbols, designating respec These socalled the numbers 7, 1000, 6, 100, 80, 5. tively to have been, like are supposed originally Singhalesian signs
the old
Hindu numerals, the
initial
letters
of
the
corre
1
Unlike the English, the first sponding numeral adjectives. nine Sanscrit numeral adjectives have each a different begin In course of centuries the ning, thereby excluding ambiguity. forms of the Hindu
letters altered materially,
but the letters
that seem to resemble most closely the apices of Boethius and the WestArabic numerals (which we shall encounter later) are
the letters of the second century. The earliest known form of the Indian symbol for zero was that of a dot, which was used in inscriptions and manuscripts in order to mark a blank. This early use resembles the still earlier use
made
of symbols for zero
by the Babylonians and
Probably Aryabhatta in the fifth century
by Ptolemy. knew our zero.
The
earliest
,A.I>.
undoubted occurrence of our
zero in India is in 876 A.D.
The Hindus possessed several different modes of designating They sometimes found it convenient to use a symbolic
numbers.
/'
1
2
in which 1 might be expressed bjj* mgon'L 2 by " eye," etc. In the tiuryctrsiddhanta 2 (a text
CXNTOK, Vol. I., pp. 603607. Translated by E. BURGESS, and annotated Ivy
New Haven,
I860, p. 3.
W. D. WHITNEY,
NUMBERSYSTEMS AND NUMERALS
13
book of Hindu astronomy) the number 1577917828 is thus two eight given Vasu (a class of deities 8 in number) mountain (the 7 mythical chains of mountains) form :
figure
(the
9
digits)
mountain
seven
which there are 15 in the halfmonth). It seems to
tainly interesting.
lunar days (of This notation is cer
have been applied as a memoria and numbers. Such a selec
technica in order to record dates
synonyms made
much
draw up phrases or a limited degree this idea may perhaps be advantageously applied by the teacher in the schoolroom. tion of
obscure verses for
it
artificial
easier to
memory. To
The Hindu notation, in its developed form, reached Europe during the twelfth century. It was transmitted to the Occi dent through the Arabs, hence the name " Arabic notation." blame attaches to the Arabs for
jSTo
this
pseudoname; they
always acknowledged the notation as an inheritance from India. During the 1000 years preceding 1200 A.IX, the Hindu
numerals and notation, while in the various stages of. eylution, were carried from country to country. Exactly what these migrations were, is a problem of extreme difficulty.
Kot even the authorship of the letters of Junius has produced so much discussion. 1 The facts to be explained and harmon ized are as follows
:
When, toward the close of the last century, scholars gradually became convinced that our numerals were not of 1.
Hindu origin, the belief was widespread that Hindu numerals were essentially identical in form. Great was the surprise when a set of Arabic numerals, the socalled Grubarnumerals, was discovered, some of which Arabic, but of the Arabic and
1
Consult TEEUTLEIN, Geschichte unserer Zahlzeichen und EntwickeSIEGMUND GUNTHEB, dieselbe, Karlsruhe, 1875
lung der Ansichten uber Ziele
;
und Mesultate der neueren mathematischhistorischen
Erlangen, 1876, note 17.
.
A HISTORY OF MATHEMATICS
14
bore no resemblance whatever to the call ed
modern Hindu characters
}
De vanaga rinumerals.
Closer research showed that the numerals of the Arabs of
2.
from those of the Arabs
Bagdad
differed
to such
an extent that
it
was
at Cordova,
difficult to believe the
and
this
westerners
received the digits directly from their eastern neighbours. The WestArabic symbols were the Grubarnurnerals mentioned above.
The Arabic digits can be traced back to the tenth century. 3. The East and the West Arabs both assigned to the num " are " dustnumerals/' erals a Hindu origin. " Gubarnumerals in memory of the Brahmin practice of reckoning on tablets strewn with dust or sand.
Not
4.
less startling
was the
fact that both sets of Arabic
numerals resembled the apices of Boethius much more closely than the modern Devanagarinumerals. The Gubarnumerals in particular bore a striking resemblance to the apices. But what are the apices ? Eoman a writer of the Boethius, tT
century, wrote a geometry, in which he speaks of an Instead of abacus, which he attributes to the Pythagoreans. following the ancient practice of using pebbles on the abacus, si'" i
he employed the
Upon
apices, which were probably small cones. each of these was drawn one of the nine digits, now
These digits occur again in the body of the Boethius gives no symbol for zero.
called "apices." text.
1
we marvel
attempting to harmonize these apparently incongruous facts, scholars for a long time failed to agree on an explanation of the strange metamorphoses of IsTeed
that, in
the numerals, or the course of their fleeting footsteps, as they migrated from land to land ?
The explanation most favourably received is that of Woepcke.* 1
2
See FRIEDLEIN'S edition of BOETHIUS, Leipzig, 1867, p. 397. See Journal Asiatique, 1st halfyear, 1863, pp. 6979 and 514629,
See also CANTOR,
VoL
I, p. 711.
NUMBERSYSTEMS AST> NUMERALS 1.
15
The Hindus possessed the nine numerals without the
zero,
known
that
as early as the second century after Christ.
It is
about that time a lively commercial intercourse was carried on between India and Borne, by way of Alexandria. There
The arose an interchange of ideas as well as of merchandise. Hindus caught glimpses of Greek thought, and the Alexan drians received ideas on philosophy and science from the East. 2.
The nine numerals, without the
zero,
thus found their
where they may have attracted the atten way tion of the XeoPythagoreans. From Alexandria they spread to Alexandria,
Home, thence to Spain and the western part of Africa, [While the geometry of Boethius (unless the passage relating to the apices be considered an interpolation made five or six to
centuries after Boethius) proves the presence of the digits in Rome in the fifth century, it must be remarked against this part of TVoepcke's theory, that he possesses no satisfactory
evidence that they were
known
in Alexandria in the second
or third century.] 3. Between the second and the eighth centuries the nine characters in India underwent changes in shape. A prominent
Arabic writer, Albir&ni (died 1038), who was in India dur many years, remarks that the shape of Hindu numerals and letters differed in different localities and that when (in the ing
eighth century) the Hindu notation was transmitted to the Arabs, the latter selected from the various forms the most
But before the East Arabs thus received the nota had been perfected by the invention of the zero and
suitable. tion,
it
the application of the principle of position. 4. Perceiving the great utility of that Columbusegg, the zero, the West Arabs borrowed this epochmaking symbol from those in the East, but retained the old forms of the nine
num
which they had previously received from Rome. The reason for this retention may have beesi a disinclination to erals,
s!l
ON
hr
*m\ ^ ^s
00
oo
bo
2
N 3
.3
I
E
s 2
1
*!
1

S !??! fllll 3
"5U
fl
g
I
1
e
P
i
1
1 *
2 S
i
2 ^9
s s o {*
fi t>
5
8"
Sfe'
IE
*
S ^
S
S ,
^
!
*:
,'" ss S, c\S ir *s ^ _x
iiI
1I1I I
* 16
i
111 If ?
i
^
'
BI
K o
5 O
_
2=s
cJ
~
ll
.
5ll
K
rt
^
lltl!
3
II 53
1
jll^j OO
hj
= ^
.2 e
I <
NUMBERSYSTEMS AND NUMERALS
17
unnecessary change, coupled, perhaps, with a desire to be con trary to their political enemies in the East. 5. The West Arabs remembered the Hindu origin of the old forms, and called them " Gubar " or " dust " numerals. 6. After the eighth century the numerals in India under went further changes, and assumed the greatly modified forms 2>f
the modern Devanagarinumerals, The controversy regarding the origin and transmission ,
,
of
our numerals has engaged many minds. Search for infor mation has led to the close consideration of intellectual,
commercial, and political conditions among the Hindus, Alex andrian Greeks, Romans, and particularly among the East and
West
We
Arabs.
have here an excellent illustration of
how
mathematicohistorical questions may give great stimulus to the study of the history of civilization, and may throw new light
upon
it.
1
In the history of the art of counting, the teacher finds em phasized certain pedagogical precepts. We have seen how universal has been the practice of counting on fingers. In place of
fingers/ 'groups
of
when South Sea
chosen, "as
other objects were frequently Islanders count with cocoanut
one aside every time they count to 10, one when they come to 100, or when African negroes reckon with pebbles or nuts, and every time they come to 5 put them aside in a little heap." 2 The abstract notion of stalks, putting a little
and a
large'
number Is here attained through the agency of concrete objects. The arithmetical truths that 2 + 1 = 3, etc., are here called out by experience in the manipulation of things. As we shall see later, the counting by groups of objects in early times led to the invention of the abacus, which is still a valuable school
instrument.
'*
GUKTHEE,
The
earliest arithmetical
op. eft., p. IS.
*
TYLOR,
knowledge of a op.
cit.,
Vol.
I.,
child
p. 270.
A HISTORY OF MATHEMATICS
18
should, therefore, be
made
to
different groups of objects
ing his
1
grow out
of his experience with
never should he be taught count
;
by being removed from his toys, and (practically with eyes closed) made to memorize the abstract statements
+ 1 = 2, 2+1 = 3,
Primitive counting in evolution emphasizes the value of objectteaching. etc.
Recent research 1 seems local value
and the use
earlier
credit of this achievement belongs to the
America.
mode
of
to indicate that the principle of
of the zero received systematic
development in America several centuries
The
its
and
full
than in Asia.
Maya
of Central
About the beginning of the Christian era the Maya
possessed a fully developed numbersystem and chronology. Theirs was not the decimal, but the vigesimal notation. Their
system, as exhibited in their codices, was based on the scale of 20, except in the third position*
That
is,
20 units of the
lowest order (kins, or days) make 1 unit of next higher order (uinals, or 20 days), 18 uinals make 1 tun (360 days, the Maya official
make 1
year), 20 tuns cycZe,
make
1 Jcatun (7200 days), 20 katuns and 20 cycles make 1 great cycle. The numbers
119 are expressed by bars and dots. Eaoii bar stands for each dot for 1. Thus, is 2, ^ is 6. The zero is
5,
repre
sented by a symbol that looks roughly like a halfclosed eye. 37 was Accordingly, expressed by the symbols for 17 (three bars and two dots) in the kin place, and one dot representing 20 in the uinal place, higher up. The numbers were written vertically.
To write
360, the
Maya drew two
zeros,
one above
the other, with a dot in third place, above the zeraa. largest number found in the codices is 12,489,781. 1 S.
G. MOELET, Introduction WasMngton, 1915.
to the
Study of
the
Maya
The
Hieroglyphs,
ARITHMETIC AND ALGEBRA
EGYPT THE most time
ancient mathematical handbook
known
to oui
a papyrus included in the Rhind collection of the Museum. This interesting hieratic document, de
is
British
1
by Birch in 1868, and translated by Eisenlohr in written by an Egyptian, Ahmes by name, in the was 1877, It is of Earaus, some time between 1700 and 2000 B.C. reign scribed
entitled
:
things."
" Directions for obtaining the knowledge of It claims to be
all
dark
founded on older documents pre
pared in the time of King [Raenm]at. Unless specialists are in error regarding the name of this last king, Eaenmat, [i.e.
Amenemhat
papyrus,
it
III.],
whose name
follows that the original
not
is is
many
legible
in the
centuries older
than the copy made by Ahmes. The Ahmes papyrus, there fore,, gives us an idea of Egyptian geometry, arithmetic, and algebra as
it
existed certainly
I ossibly as early as 3000 B.C.
*
as
early
While
it
as
1700
B.C.,
and
does not disclose as
extensive mathematical knowledge as one might expect to find among the builders of the pyramids, it nevertheless shows in 1 A. EISESXOHR, Bin mathematisches Handbuch der alten Aegypter (Papyrus EMnd des British Museum), Leipzig, 1877; 2d ed. 1891. See also CANTOR, L, pp. 5984, and JAMES Gow, A Short History of Greek Mathematics, Cambridge, 1884, pp. 1619. The last work will be cited
hereafter as
Gow. id
A HISTORY OF MATHEMATICS
20
a remarkably advanced state of mathe,
several particulars
when Abraham visited Egypt. From the Ahmes papyrus we infer that the Egyptians knew
matics at the time
nothingjyP
t'hAnratif'.a'l
mm
no theorems, and In most cases the
It contains
Its.
hardly any general rules of procedure.
writer treats in succession several problems of the same kind.
From them rules,
it
but this
would be easy, by induction, to obtain general not done. When we remember that only one
is
hundred years ago
it was the practice of many English arith metical writers to postpone the discussion of fractions to the end of their books, it is surprising to find that this handbook
of 4000 years ago begins with exercises in fractions, and pays but little attention to whole numbers. Gow probably is right in his conjecture that Ahmes wrote for the 4lite mathematicians
of his day.
While
fractions occur in the oldest mathematical records
which have been found, the ancients nevertheless attained little proficiency in them. Evidently this subject was one of Simultaneous changes in both numerator and denominator were usually avoided. Fractions are found among theBabvlonians. ISFot only had they sexagesimal divisions of great difficulty.
weijipts *fr3erions
and measures, but also sexagesimal fractions. 1 These had a constant denominator (60), and were indicated
by writing the numerator a little to the right of the ordinary 'position for a word or number, the denominator being under
We
shall see that the Eomans likewise usually k.pt the denominators constant, but equal to 12. The Egyptians and Greeks, on the other hand, kept the numerators constant
stood,
,
ancLdgajt with variable denominators. Thr^s 'confines self to .fractions of a special class, namely unitfractions,
Em
having
unity for their numerators. 1
A
CAUTOB, Vol.
I.,
fraction
was designated by
pp. 23, 31.
EGYPT
21
writing the denominator and then placing over it either a dot ro. Fractional values which could not be
or a symbol, called
expressed by any one unitfraction were represented by the sum of two or more of them. Thus he wrote in place of f.
^
It
curious to observe that while
is
he makes an exception in
to equal
this case, adopts a special

fa
symbol
1 appear often among the unitfractions. fundamental problem in Ahmes's treatment of fractions was,
for f ,
A
and allows
Ahmes knows f
how
it to
to find the unitfractions, the
sum
which represents a
of
This was done by aid of a table, given
given fractional value.
f\
in the papyrus, in
which
all
fractions of the

form 2ft
(n designating successively the sum of unitfractions.
With
all
integers
Thus,
up
+l
to 49) are reduced to
^^^^^
^ = ^ ^,
Ahmes
could work out problems like these, "Divide 2 by 3," "Divide 2 by 17," etc. Ahmes nowhere states why he confines himself to 2 as the numer grjy.
aid of this table
nor does he inform us how, when, and by
ator,
was constructed.
It is
plain, however, that
whom
the table
by the use of
any fraction whose denominator is odd and less than can be 100, represented as the sum of unitfractions. The division of 5 by 21 may have been accomplished as follows this table
:
5
= 1+2 + 2.
From
the table
we
get
^=
&
^.
Then
=
I ~h A Jt ma7 be remarked that there are many ways of breaking up a fraction into unitfractions, but ATvm.es invariably Contrary to Ms usual practice, he gives a " If rule for general multiplying a fraction by . He says are what is of you asked, f fa take the double and the sixfold gives only one.
:
;
that
is
tion."
 of
One must proceed likewise
any other frac As only the denominator was written down, he means it.
,
Vol. I.,p/61.
for
A HISTORY OF MATHEMATICS
22
" " " by the double and sixfold," double and sixfold the denon> Since f = 1, the rule simply means inator. J
2
1
3
5
=
1
1
2x56x5*
His statement "likewise for any other fraction" appears
to
mean 1 a~~2a
3
6a
The papyrus contains 17 examples which show by what a fraction or a mixed number must be multiplied, or what must be added to
it,
to obtain a given value.
The method
in the reduction of the given fractions to a
consists
common denomina
Strange to say, the latter is not always chosen so as to be exactly divisible by all the given denominators. A times The common gives the example, increase i i iV TF *V fco 1denominator taken appears to be 45, for the numbers are
tor.
stated as 11 J, 5J
J,
fortyfifths.
Add
and we have
1.
4J,
1.
1^,
The sum
^, and
to this
Hence the quantity
of these
the to be
sum
is
is
23  J
Add
f.
added
J,
to the given
fraction is
^ ^. By what must
^ ^ be
mon denominator O
= 9' then
taken "J
Again
Q
Ql
= oi'
multiplied to give  ?
The com
then T^ = TO^J TTO = i, their ^^ 11^ 2o Since 2 + H i = 3^, take first A,
is 28,
*^
Hence ^, then half of that' /o and we have ft ^o ^^co^es on iV dhr multiplication by 1 J. These examples disclose methods quite foreign to modern mathematics. 2 One process, however, found extensive appli
J.
its
half

,
1
GAKTOR, Vol.
2
Cantor's account of fractions in the
I.,
p. 66.
Ahmes papyrus
Sylvester (then in Baltimore) material for a paper "
the Theory of Yulgar Fractions," pp. 32 and 388.
suggested to a Point in
On
Am. Journal of Mathematics,
III., 1880.
EGYPT cation in
arithmetics
of
the
23 century and
fifteenth
later,
namely, that of aliquot parts, used largely in Practice. In the second of the above examples aliquot parts are taken of fa.
This process
seen again in Ahmes's calculations to verify
is
the identities in the table of unitfractions. .A limes then proceeds to eleven problems leading to simple equations with one unknown quantity. The unknown is called hau or 'heap.' Symbols are used to designate addition, sub
traction,
A
I
1
1
Ha
and
;
its
&
Here
f
,
maf its J,
zQ
2.6.
give the following specimen
* :
^ni
ir nebf
Heap
We
equality.
stands for
f
,
indicated by writing the
J /
ro sefe^f hif x eP erf its I , its whole, it gives f
for .
37
=
f 1)
i
em 37
Other unitfractions are
number and placing C3^,
ro, above it. problem resembling the one just given reads: "Heap, its , " i.e. x its f, its whole, it makes 33 its +  f ^ f x = 33. 3 The solution proceeds as follows 1 f  $ # = 33. Then 1 1 ^
A
,
27
;
:

multiplied in the manner sketched above, so as to get 33,
is
thereby obtaining heap equal to 14 \ fc JF ^g y^g T J T ^.. Here, then, we have the solution of an algebraic equation !
In this Egyptian document, as also among early Babylonian records, are found examples of arithmetical and geometrical Ahmes gives an example " Divide 100 loaves progressions. :
among two
2
\ of what the
first
three get
is
what the
What is the difference ? V Ahmes gives "Make the difference 5; 23, 17J, 12, 6, 1.
LUDWIG MATTHIESSEX, Grundzuge
bra der this
;
get.
tion: 1
5 persons
litteralen Grleichungen,
work
as MATTHIESSEN.
CANTOK, Vol. L,
p. 78.
der Antiken
last
the solu
Multiply
und Modernen Alge
Leipzig, 1878, p. 269.
Hereafter
we
cite
A HISTORY OF MATHEMATICS
24
How did Ahmes come upon If." l Let a and d be the first term and thus Perhaps the difference in the required arithmetical progression, then by If
;
38J, 291, 20, 10
5^ ?
 [a
,
:

+ (a  d) +
5J(a
4d),
i.e.
Assuming the
sum is 60 x If
60,
last
but
= 100.

=
=
(a (a 2d)] 3d) 4 (a 4cZ), whence d the difference d is 5J times the last term.
term
= 1,
he gets his
first
progression.
The
should be 100; hence multiply by If, for We have here a method of solution which
appears again later among the Hindus, Arabs, and modern the method of false position. It will be ex Europeans plained more fully elsewhere. Still
more curious
is
the following in Ahmes.
of a ladder consisting of the
numbers
7, 49,
He
speaks
343, 2401, 16807.
Adjacent to these powers of 7 are the words picture, cat, mouse, barley, measure. Nothing in the papyrus gives a clue to the meaning of this, but Cantor thinks the key to be found in the following problem occurring 3000 years later in the liber abaci 7 old women go to Rome (1202 A.D.) of Leonardo of Pisa :
;
each
woman
has 7 mules, each mule carries 7 sacks, each sack contains 7 loaves, with each loaf are 7 knives, each knife is
put in 7 sheaths. What is the sum total of all named ? This has suggested the following wording in Ahmes: 7 persons have each 7 cats ; each cat eats 7 mice, each mouse eats 7 ears of barley,
from
each ear 7 measures of corn may grow. What the series arising from these data, what the sum of its terms? Ahmes gives the numbers, also their sum, 19607. Problems of this sort may have been for amusement. is
proposed
If the above interpretations are correct,
matical
recreations"
were
indulged
it
in
looks as
by
if
"
scholars
mathe forty
centuries ago.
In the famproblems, of which we gave one example, we see the beginnings of algebra. So far as documentary evidence 1
CANTOR, Yol. L,
p. 78.
EGYPT
25
and algebra are coeval. That arithmetic is But mark the older there can be no doubt. the actually close relation between them at the very beginning of authentic goes, arithmetic
So in mathematical teaching there ought to be an intimate union between the two. In the United States algebra
history.
was for a long time set aside, while extraordinary emphasis was laid on arithmetic. The readjustment has come. The "Mathematical Conference of Ten" of 1892 voiced the senti ment of the best educators when it recommended the earlier introduction of certain parts of elementary algebra. The part of Ahmes's papyrus which has to the
greatest
degree taxed the ingenuity of specialists is the table of unitfractions. How was it constructed ? Some hold that it was
not computed by any one person, nor even in one single epoch, and that the method of construction was not the same for all
On
fractions.
the other hand Loria thinks he has discovered
a general mode by which this and similar existing tables
may
have been calculated. 1
That the period of Ahmes was a flowering time for Egyptian mathematics appears from the fact that there exist two other papyri of this period, containing quadratic equations were found at Kahun, south of the pyramid of Illahun.
!
They These
documents bear
Akhmim
close resemblance to Ahmes's, as does also the 2 papyrus, recently discovered at Akhiaim, a city on
1 See the following articles by GINO LORIA: "Congetture e ricerche sulP aritmetica degli antichi Egiziani " in Bibliotheca Mathematics 1892,
miovo documents relative alia logistieo grecoegiziana, " Ibid., 1893, pp. 7989; "Studi intorno alia logistica grecoegiziana," Estratto dal Volume XXXII (1 della 2 a serie) del Giornale di Mathemapp. 97109
;
"Un
tiche di Battaglini, pp. 735. 2
J.
BAILLET,
"Le
papyrus mathe'matique d'Akhmim," Memoires pub
membres
de la mission archeologique frangaise au Caire, See also LORIA'S papers in the preceding note, and CANTOR, Vol. I., 1907, pp. 95, 96.
lies
T.
par
JX,
les
l r fascicule, Paris, 1892, pp. 188.
A HISTOEY OF MATHEMATICS
26 the Nile in
Greek and is supposed to some time between 500 and 800 A.D. Like
Upper Egypt.
have been written at
It
is
in
author gives tables of unitIt marks no progress over the arithmetic of Ahmes. fractions. For more than two thousand years Egyptian mathematics was
his ancient predecessor
stationary
Ahmes,
its
I
GEEECE Greek arithmetic and algebra, we first observe that the early Greeks were not automaths they acknowledged the Egyptian priests to have been their teachers. While in geometry the Greeks soon reached a height undreamed of by In passing
to
;
the Egyptian mind, they contributed hardly anything to the Not until the golden period of geometric art of calculation.
and discovery had passed away, do we find in Nicomachus to algebra. Diophantus substantial contributors Greek mathematicians were in the habit of discriminating between the science of numbers and the art of computation.
The former they
called arithmetica, the latter logistica.
Greek writers seldom refer to calculation with alphabetic numerals. Addition, subtraction, and even multiplication were Eutocius, a commentator probably performed on the abacus. a exhibits the sixth of great many multiplications, century A.D., such as expert Greek mathematicians of classical time may
have used. 1
some
While among the Sophists computation received it was pronounced a vulgar and childish
attention,
by Plato, who cared only for the philosophy of arithmetic. Greek writers did not confine themselves to unitfractions as Unitfractions were designated closely as did the Egyptians. art
by simply writing the denominator with a double 1
accent.
specimens of such multiplications see CANTOR, JBeitrage z. JTulturl. HANKEL, p. 56 Gow, p. 50 ; FRIEDLEIN, p. 76 ; my History of Mathematics, 1895, p. 65. d.
3?or
Volker, p. 393
;
;
GBEECE
27
=
yf 2. Other fractions were usually indicated by once with an accent and the denomina numerator the writing tor twice with a double accent. Thus, i J. As Thus,
pt/3"
WW =
with the Egyptians; unitfractions in juxtaposition are to be added.
Like the Eastern nations, the Egyptians and Greeks employ
two aids is
not
to computation, the abacus
known what
and
finger symbolism.
If
the signs used in the latter were, but by the
study of ancient statuary, basreliefs, and paintings this secret may yet be unravelled. Of the abacus there existed many
forms at different times and among the various nations. In all cases a plane was divided into regions and a pebble or other
We
object represented a different value in different regions. possess no detailed information regarding the Egyptian or the Greek abacus. Herodotus (II., 36) says that the Egyp tians " calculate with pebbles
by moving the hand from right move it from left to right." This indicates a primitive and instrumental mode of counting with aid of pebbles. The fact that the hand was moved towards the to left, while the Hellenes
right, or
towards the
left,
indicates that the plane or board
was
divided by lines which were vertical, i.e. up and down, with lamblichus informs us that the respect to the computer.
abacus of the Pythagoreans was a board strewn with dust or In that case, any writing could be easily erased by sand. sprinkling the board anew.
space or
column designated
A pebble placed in the right hand 1, if
placed in the second column
from the right 10, if in the third column 100, etc. Probably, never more than nine pebbles were placed in one column, for ten of them would equal one unit of the next higher order. The Egyptians on the other hand chose the column on the extreme
left as the place for units, the
left designating tens, the
second column from the
third hundreds,
etc.
In further
support of this description of the abacus, a comparison attritn
A HISTOKT OF MATHEMATICS
28
uted by Diogenes Laertius (I., 59) to Solon is interesting l "A person friendly with tyrants is like the stone in computation, :
which
signifies
now much, now
little."
have been used in Egypt and Greece The hand to carry out the simpler calculation with integers. book of Ahmes with its treatment of fractions was presumably
The abacus appears
to
written for those already familiar with abacal or digital reck Greek mathematical works usually give the numerical oning.
without exhibiting the computation itself. Thus advanced mathematicians frequently had occasion to extract the square root. In his Mensuration of the Circle, Archimedes
results
states, for instance,
that
V3 < ^g^
1
and
V3 > fff
,
but he
method of approximation. 2 When sexagesimal numbers (introduced from Babylonia into Greece about the time of the Greek geometer Hypsicles and the Alexandrian astronomer Ptolemseus) were used, then gives no clue to his
the mode of rootextraction resembled that of the present time. specimen of the process, as given by Theon, the father of
A
=
67 4' 55". Hypatia, has been preserved. He finds V4500 Archimedes showed how the Greek system of numeration
might be extended so
as to
embrace numbers
as large as
you
the ordinary nomenclature of his day, numbers s could be expressed up to 10 8 Taking this 10 as a unit of
please.
By
.
second order, 10 16 as one of the third order, etc., the system may be sufficiently extended to enable one to count the very
Assuming 10,000 grains of sand to fill the space of a poppyseed, he finds a number which would exceed the number sands.
1 2
CAKTOE, Vol. I., p. 132. What the Archimedean and, in general, the Greek method of
root
extraction really was, has "been a favourite subject of conjecture. See for example, H. WEISSENBOEN'S Berechnung des JKreisumfanges bei Archi
medes und Leonardo Pisano, Berlin, 1894. ject,
For bibliography of this sub see S. GtJNTHEE, Gesch. d. antiken Naturwissenschaft u. Philosophic,
p. 16.
GREECE whose radius extends from the earth
of grains in a sphere
the fixed
A counterpart
stars.
29 to
of this interesting speculation,
" called the " sandcounter (arenarius), is found in a calcula tion attributed to Buddha, the Hindu reformer, of the number
of primary atoms in a line one mile in length,
when
the atoms
are placed one against another. The science of numbers, as distinguished from the art of
commanded the
calculation,
lively attention of the
Pythago
Pythagoras himself had imbibed Egyptian mathemat Aside from the capital discovery of ics and mysticism. reans.
irrational quantities (spoken of elsewhere)
no very substantial
made by the Pythagoreans to the science of add that by the Greeks irrationals were The Pythagoreans sought the not classified as numbers. in numbers all of harmony depended on musi things origin cal proportion; the order and beauty of the universe have contribution was
We may
numbers.
;
their origin in
numbers
cerned a wonderful
;
in the planetary motions they dis
"harmony
of
the spheres."
Moreover,
some numbers had extraordinary attributes. Thus, one is the essence of things four is the most perfect number, correspond ;
ing to the
human
soul.
According to Philolaus, 5 is the cause and light, 8 of love and
of colour, 6 of cold, 7 of mind, health,
Even Plato and Aristotle refer the virtues to While these speculations in themselves were fan and barren, lines of fruitful mathematical inquiry were 1
friendship.
numbers. tastic
suggested by them.
numbers into odd and even, and observed that the sum of the odd numbers from 1 to 2 n f 1 was always a perfect square. Of no particular value were their classifications of numbers into heteromecic, trian
The Pythagoreans
classified
gular, perfect, excessive, defective, amicable. 1 2
Gow, p. 69. For their definitions see Gqw,
2
p. 70, or CAJOEI, op.
The Pythagcit,, p. 68.
A HISTOEY OF MATHEMATICS
30 oreans paid
much
attention to the subject of proportion.
Tha,,
quantities a, b, c, d, were said to be in arithmetical proportion, when a b=cd; in geometrical proportion, when a b = c d :
:
;
in har/nojiic proportion,
when a
"proportion,
:

when a (a f &)
&:&
c=a:c;
= 2 ab/(a + b):b.
in musical
lamblichus
says that the last was introduced from Babylon. The 7th, 8th, and 9th books of Euclid's Elements are on the science of number, but the
2d and 10th, though professedly
geometrical and treating of magnitudes, are applicable to numbers. Euclid was a geometer through and through, and even his arithmetical books smack of geometry. Consider, for 1 " Plane and solid numbers instance, definition 21, book VII. are similar when their sides are proportional." Again, num bers are not written in numerals, nor are they designated by
anything like our modern algebraic notation they are repre sented by lines. This symbolism is very unsuggestive. Fre quently properties which our notation unmasks at once could ;
be e^traMed from these lines only through a severe process of 2
reasoning.
In the 7th book we encounter for the of prime numbers.
first
time a definition
Euclid finds the GfTe. P. of two numbers
by a procedure identical with our method by division. He applies to numbers the theory of proportiojUwhich in the 5th book is developed for magnitudes in general. The 8th book deals with
numbers
in continued proportion.
The 9th book
finishes that subject, deals
with primes, and contains the proof for the remarkable theorem 20, that the number of primes is infinite.
During the four centuries after Euclid, geometry monopo Greeks and the theory of numbers
lized the attention of the 1
2
To
EEIBERG'S
edition, Vol. II., p. 189.
G. E. F. NESSELMANN, Die Algebra der Griechen, Berlin, 1842, p. 184, be cited hereafter as NESSELMANN.
GREECE
31
Of this period only two names deserve men Eratosthenes (about 275194 B.C.) and Hypsicles (between tion, 200 and 100 B.C.). To the latter we owe researches on polyg was neglected.
onal numbers and arithmetical progressions. Eratosthenes invented the celebrated "sieve" for finding prime numbers.
Write down in succession
all odd numbers from 3 up. By number after 3, sift out all multiples of 3 by erasing every fifth number after o, sift out all multiples The numbers left after this sifting are all of 5, and so on.
erasing every third
prime.
;
"While the invention of the " sieve " called for no
great mental powers,
it is
remarkable that after Eratosthenes
no advance was made in the mode of finding the primes, nor in the determination of the number of primes which exist in the numerical series
1,
2, 3,
...
the nineteenth century,
n, until
when
Gauss, Legendre, Dirichlet, Eiemann, and Chebichev enriched the subject with investigations mostly of great diffi
culty and complexity. The study of arithmetic was revived about 100 A.D. by 1 Nicomachus, a native of Gerasa (perhaps a town in Arabia)
and known as a Pythagorean.
He
entitled Introductio Arithmetica.
The
this
work
is great,
not so
wrote in Greek a work
historical importance of account of original matter is (so far as we know) the
much on
therein contained, but because earliest systematic textbook
it
on arithmetic, and because
for
over 1000 years it set the fashion for the treatment of this subject in Europe. In a small measure, Nicomachus did for arithmetic
what Euclid did
for geometry.
His arithmetic was
famous in his day as was, later, Adam Biese's in Germany, and Cocker's in England. Wishing to compliment a computor, Lucian says, " You reckon like Nicomachus of Gerasa." 2 The as
1
See NESSELMAXN, pp. 191216
;
Gow,
pp. 8895
pp. 400404. 2
Quoted by
Gow
(p. 89)
from Philopatris,
12.
;
CJLNTOB, Yol. I,
A HISTOKY OF MATHEMATICS
32
work was brought out in a Latin translation by Appuleius (now lost) and then by Boethius. In Boethius's translation the elementary parts of the work were in high authority in Western Europe until the country was invaded by Hindu Thereupon for several centuries Greek arithmetic bravely but vainly struggled for existence against its immeas arithmetic.
urably superior Indian
The
style of
rival.
Mcomachus
from that of his
differs essentially
The geo predecessors. metrical style is abandoned the different classes of numbers are exhibited in actual numerals. The author's main object is It is not deductive, but inductive. ;
Being under the influence of philosophy and theology , he sometimes strains a point to secure a division into groups of three. Thus, odd numbers are either "prime and
classification.
nncompounded," "compounded," or "compounded but prime to one another." His nomenclature resulting from this classi fication is exceedingly
for his
burdensome.
The Latin
equivalents of his
Greek terms are found in the printed arithmetics
disciples,
1500 years
Thus the
later.
l
particularism
m 4 1
is
m+
ratio
subsuperparticularis,
3J
=
is
superis
4 in of numbers Mcomachus tables gives triplex sesquiquartus. form of a chessboard of 100 squares. It might have answered 1
as a multiplicationtable, but
the study of ratios. 2
He
it
appears to have been used in
describes polygonal numbers, the
different kinds of proportions (11 in all),
summation of numerical
series.
rules of computation, of problemworking,
arithmetic.
He
and
To be noted
is
treats of the
the absence of
and
of practical
gives the following important proposition.
All cubical numbers are equal to the 1
Gow,
2
FBIEDLEIN, p. 78
sum
of successive odd
pp. 90, 91. ;
CANTOR, Vol. L,
p. 431.
GREECE
= 2 = 3 + 5; + 19.
numbers.
4
s
Thus, 8 15 + 17
= 13 +
3
33
27
= S = 7 + 9 + 11 3
;
64
=
Kicomachus, lamblichus, Theon of and others, are found investigations Smyrna, Thymaridas, in their nature. algebraic Thymaridas in one place uses a Greek word meaning " unknown quantity " in a manner sug
In the
writings
of
Of interest in tracing gesting the near approach of algebra. the evolution of algebra are the arithmetical epigrams in the' Palatine Anthology, which contained about 50 problems leading to linear equations. 1
Before the introduction of algebra, these problems were propounded as puzzles. "No. 23 gives the times in which four fountains can fill a reservoir separately and 2 Ko. 9. What requires the time they can fill it conjointly. part of the day has disappeared, if the time left is twice two
thirds of the time passed away ? Sometimes included among these epigrams is the famous "cattleproblem," which Archi
medes
is
said to have propounded to the Alexandrian
mathe
3
In the
maticians.
This
difficult
problem
is
indeterminate.
part of it, from only seven equations, eight unknown Gow states quantities in integral numbers are to be found. it thus: The sun had a herd of bulls and cows, of different first
Of Bulls, the white ( W) were in number ( \the B were (J f ) of the 7 (J5) and yellow (F) and Y. (2) Of of the and piebald (P) the P were ( + Cows, which had the same colours (w, b, y,p)> w ( colours.
(1)
J)
of the blue
;
}
W
1 These epigrams were written in Greek, perhaps about the time of Constantine the Great. For a German translation, see G. WEETHEIM,
Die Arithmetik und die Schrift uber PolygonalzahUn des Diophantus von Alexandria, Leipzig, 1890, pp. 330344. 2
WERTHEIM, op. cit., p. 337. Whether it originated at the time of Archimedes or later is discussed by T. L. HEATH, Diophantos of Alexandria, Cambridge, 1885, pp. 1428
147.
A HISTORY
34
Find the number of
bulls
OF MATHEMATICS
and cows.
high numbers, but to add to
its
This leads to excessively complexity, a second series of
superadded, leading to an indeterminate equa tion of the second degree. conditions
is
Most of the problems in the Palatine Anthology, though Such puzzling to an arithmetician, are easy to an algebraist. of and became about the time Diophantus problems popular doubtless acted as a powerful mental stimulus. Diophantus, one of the last Alexandrian mathematicians,
generally regarded as
died about 330 A.D.
is
an algebraist of great fertility. 1 He His age was 84, as is known from an
epitaph to the following effect: Diophantus passed  of his childhood, ^ in youth, and ^ more as a bachelor five
life in
;
years after his marriage,
was born a son who died four years
before his father, at half his father's age. This epitaph states about all we know of Diophantus. are uncertain as to the
We
time of his death and ignorant of his parentage and place of Were his works not written in Greek, no one would nativity. suspect them of being the product of Greek mind. The spirit
pervading his masterpiece, the Arithmetica [said to have been written in thirteen books, of which only six (seven ?) 2 are ex tant] is as different from that of the great classical works of the time of Euclid as pure geometry is from pure analysis.
Among the
Greeks, Diophantus had no prominent forerunner, disciple. Except for his works, we should be
no prominent
obliged to say that the Greek mind accomplished nothing notable in algebra. Before the discovery of the Ahmes
papyrus, the Arithmetica of Diophantus was the oldest
work on
known
Diophantus introduces the notion of an algebraic equation expressed in symbols. Being completely 1
"How
algebra.
far
was Diophautos original?"
159. 2
CHITA*, Vol.
I.,
pp. 466, 467.
see
HEATH,
op. c#., pp. 133
GREECE
35
divorced from geometry, his treatment is purely analytical. He is the first to say that " a number to be subtracted multi
number
plied by a
This
added."
is
number
to be subtracted gives a
applied to differences, like (2
x
to be
3) (2 x
3),
the product of which he finds without resorting to geometry. 2 Identities like (a f 6) 2 a2 f 2 ab 5 , which are elevated
=
+
rank of geometric theorems, with Diophantus are the simplest consequences of algebraic laws of operation. Diophantus represents the unknown quantity x by
by Euclid
to the exalted
U
v
V
the square of the unknown a? by 8 x3 by * x* by SS His sign for subtraction is 41, his symbol for equality, t. Addi tion is indicated by juxtaposition. Sometimes he ignores these s',
.
,
,
symbols and describes operations in words, when the symbols In a polynomial all the positive better. terms are written before any of the negative ones. Thus,
would have answered a3
o or h S
would be in his notation, 1
1
Here the numerical
u /c
ot
^8
as ^
v
e/K,d.
coefficient follows the x.
To be emphasized is the fact that in Diophantus the funda mental algebraic conception of negative numbers is wanting. In 2 x 10 he avoids as absurd all cases where 2 x < 10. Take Probl 16 Ek. bers such that the
I.
in his Arithmetica
sums
:
"
To
find three
a, by c
are the given numbers, then one of the required
is
{
%(a
&
}
c)
c.
Ifc> lf(a j 6
f c),
must be greater than any one
singly."
give solutions general in form. special values 20, 30, 40 are
unin
problem numbers
Diophantus does not
In the present instance the
to simultaneous equations
HEATH,
is
assumed as the given numbers.
adroitly uses only one symbol for the
1
If
numbers
then this result
Hence he imposes upon the telligible to Diophantus. the limitation, " But half the sum of the three given
In problems leading
num
of each pair are given numbers."
op. cit, p. 72.
unknown
Diophantus quantities.
A HISTORY OF MATHEMATICS
36
This poverty in notation is offset in many cases merely by skill in the selection of the unknown. Frequently he follows a "
position
:
method resembling somewhat the Hindu "false a preliminary value is assigned to the unknown
only one or two of the necessary conditions. This leads to expressions palpably wrong, but nevertheless
which
satisfies
suggesting some stratagem 1 values can be obtained.
by which one
of
the
correct
Diophantus knows how to solve quadratic equations, but in the extant books of his Arithmetica he nowhere explains the mode of solution. Noteworthy is the fact that he always gives but one of the two roots, even when both roots are positive. Nor does he ever accept as an answer to a problem a quantity
which
is
negative or irrational. first book in the Arithmetica
Only the
is
devoted to deter
It is in the solution of indeterminate minate equations. the second equations (of degree) that he exhibits his wonderful
inventive faculties.
However, his extraordinary
ability lies
less in discovering general methods than in reducing all sorts
of equations to particular forms which he knows how to solve. Each of his numerous and various problems has its own distinct
method
of solution,
mathematician,
which "It
related problem.
after
to solve the 101st.
.
is
is,
often useless in the most closely
therefore,
studying .
.
difficult
for a
100 Diophantine
modern
solutions,
Diophantus dazzles more than he
2
delights."
The absence
in Diophantus of general methods for dealing
with indeterminate problems compelled modern workers on this subject, such as Euler, Lagrange, Gauss, to begin anew. 1
2
Gow, pp. HANKEL,
110, 116, 117. p.
165.
It
should be remarked that HEATH, op.
83120, takes exception to HaiikePs verdict
general account of Diophantine methods.
cit.,
pp.
and endeavours to give a
KOME
37
Diophantus could teach them nothing in the way of general methods. The result is that the modern theory of numbers is quite distinct and a decidedly higher and nobler science than
Diophantine Analysis. Modern disciples of Diophantus usually display the weaknesses of their master and for that reason
have failed to make substantial contributions to the subject. Of special interest to us is the method followed by Diophan tus in solving a linear determinate equation. His directions are " If now in any problem the same powers of the unknown :
occur on both sides of the equation, but with different coeffi cients, we must subtract equals from equals until we have one term^equal to one term. If there are on one side, or on both sides,
terms with negative
added on both tive terms.
coefficients,
until there remains only one
what
these terms must be
on both sides there are only posi Then we must again subtract equals from equals sides, so that
term in each number."
1
Thus
nowadays achieved by transposing, simplifying, and dividing by the coefficient of x was accomplished by Diophan It is to be observed that in tus by addition and subtraction. is
}
Diophantus, and in fact in all writings of antiquity, the con ception of a quotient is wanting. An operation of division is
nowhere exhibited. another, the answer
When
one number had to be divided by
was reached by repeated
subtractions. 2
EOME Of Eoman methods
of computation more is known than of Abacal reckoning was taught in schools. Writers refer to pebbles and a dustcovered abacus, ruled
Greek or Egyptian.
into columns.
An
Etruscan
(?) relic,
shows a computer holding in his 1
WEETHEIM'S Diophantus^
3
$SSELMA2TN',
p.
112
J
now
left
preserved in Paris,
hand an abacus with
p. 7.
FfitEDLEIN, p. 79.
A HISTORY OF MATHEMATICS
38
hand he lays numerals set in columns, while with the right pebbles upon the table.
1
also another
The Romans used
kind of abacus, consisting
of
buttons. By its a metallic plate having grooves with movable some as frac well as use all integers between 1 and 9,999,999, two In the adjoining figures could be represented. tions,
21 in Friedlein) the lines represent grooves (taken from Pig.
l!
and the
X 7
C
X
'X
1
circles buttons.
The Roman numerals
indicate the
value of each button in the corresponding groove below, the button in the shorter groove above having a fivefold value.
= 1,000,000
hence each button in the long lefthand for 1,000,000, and the button in the groove, when in use, stands short upper groove stands for 5,000,000. The same holds for the other grooves labelled by Eoman numerals. The eighth
Thus
11
;
5 buttons) represents duo long groove from the left (having while the button decimal fractions, each button indicating the upper ninth column the In means dot the above f$.
^
^
the middle ^, and two lower each fa. Our first figure represents the positions of the buttons before the operation begins our second figure stands for the number The eye has here to distinguish the buttons in use
button represents
;
852^3^. and those
Those counted are one button above
left idle. 1
CANTOR, Vol.
I.,
p. 529.
KOME c
(=
39
and three buttons below c (= 300)
500),
one button
;
above x (=50); two buttons below I (=2); four buttons and the button for fa. indicating duodecimals (= )
;
Suppose now that 10,318 J $
^
be added to 852 $fa> The operator could begin with the highest units, or the lowest units, as he pleased. Naturally the hardest part is the addi
In
tion of the fractions.
is to
this case the button for
^
the button
above the dot and three buttons below the dot were used to
^
sum f The addition of 8 would bring all the buttons abovTahd beiow 1 into play, making 10 units. Hence, move them ill back ano^taove up one button in the groove indicate the
below
x
x.
Add
add 300
;
10 by moving up another of the buttons below by moving back all buttons above and below
to 800
except one button below, and moving up one button below I; add 10,000 by moving up one button below x. In subtraction the operation was similar. c,
Multiplication could be carried out in several ways. In case 4 fa times 25 ^the abacus may have shown successively
of 38
the
770
following
(= 760
+
values\600 
20),
770
(==
f*(=
30
770

760 (= 600
20),
+ fa

20),
+ 20
920(=
960f (= 920 fft+ 8 5), 963 (= 960 f 963 i fa (= 963 J + fa 5)Ss973 fa (= 963 f
30

5),
+
8),
770f 
*
5),
.
In division the abacus was used &a represent the remainder resulting from the subtraction from thk dividend of the divisor or of a convenient multiple of the divisor.
complicated and tion
show
difficult.
clearly
how
The
process was
These methods of abacal computa multiplication
or
diVis^pn
can
be
by a series of successive additions or sufete^ctions. In this connection we suspect that recourse was had to carried out
1
PEIEDLEIN, p. 80.
A HISTORY OF MATHEMATICS
40
operations and to the multiplication table. Fingerreckoning may also have been used. In any case the multiplication and division with large
numbers must have been beyond the power
This difficulty was sometimes the ordinary computer. of arithmetical tables from which the obviated by the use
of
required sum, difference, or product of two numbers could be Tables of this sort were prepared by Victories of copied. Aquitania, a writer who is well known for his canon paschalis,
a rule for finding the correct date for Easter, which he pub The tables of Victorius contain a peculiar
lished in 457 A.D.
notation for fractions, which continued in use throughout the
middle ages. 1
Fractions occur
among
Why
observed.
Eomans most
the
quently in money computations. The Eoman partiality to duodecimal
fractions
duodecimals and not decimals
?
is
to
fre
be
Doubtless
because the decimal division of weights and measures seemed unnatural. In everyday affairs the division of units into 2, 3, 4, 6
equal parts
is
the commonest, and duodecimal fractions
In duodecimals the give easier expressions for these parts. in decimals these of the whole above parts are fa, T%, f%
^
5
parts are
^
31 21 ,
1^
^, ^.
;
Unlike the Greeks, the Eomans dealt
The Eoman as, originally a copper coin weighing one pound, was divided into 12 uncice. The abstract fraction was expressed concretely by deunx (= de
with concrete fractions.
^
uncia,
(=
i.e.,
as [1] less
undo,
[y^])
5
thus each
A
was ca^ e(l quincunx fraction had a
Eoman
quinque [five] uncice) Addition and subtraction of such fractions were ;
special name.
easy.
Fractional computations were the chief part of arith in "Rom^n ^ryVlgJ Horace, in remem 2
metical instruction
brance, perhaps, of his
1
own
schooldays, gives the following
Consult FEIEDLEIN, pp. 9398.
2
HAITKEI,, pp. 58, 59.
41
BOMB
V. 326330) dialogue between teacher and pupil (Ars poetica, " Let the son of Albinos tell ounces five from if [i.e. ^] me, be subtracted one ounce [i.e. TV], what is the remainder? :
Come, you can
tell.
'
Good; you will be able to If one ounce [i.e. 5^] be added,
Onethird.'
take care of your property. " what does it make ? i Onehalf.'
the Romans unconsciously hit upon a fine in their concrete treatment of fractions. idea pedagogical fractions in connection with money, learned Roman boys
Doubtless
weights, and measures.
meant more " broken
We conjecture
than what was
that to
them
fractions
conveyed by the definition,
number," given in old English arithmetics.
524), who a work, is to be mentioned in this connection as the author of De Institutione Arithmetica, essentially a translation of the
One
Roman
of the last
writers
was Boethius (died
arithmetic of I^icomachus, although some of the most beau tiful arithmetical results in the original are omitted by
The
Boethius.
historical importance of
this translation lies
in the extended use made of it later in Western Europe. The Roman laws of inheritance gave rise to numerous arith metical examples. The following is of interest in itself, and also because its occurrence elsewhere at a later period assists
us in tracing the source of arithmetical knowledge in Western with child, dying man wills, that if his wife, being
Europe
:
A
shall receive f, and she  of his gives birth to a son, the son is if a but estate born, the daughter shall receive daughter ;
,
a
and happens that twins are born, a boy as to so be divided shall the estate satisfy the
and the wife f girl.
will?
How The
.
It
celebrated
Roman
jurist,
Salvianus
Julianus,
should be divided into seven equal parts, of which four should go to the son, two to the wife and one
decided that
it
to the daughter.
Aside from the (probable) improvement of the abacus and
A HISTORY OF MATHEMATICS
42
Eomans made no The algebra of Diophantus wa&j With them as with all nations of antiquity
the development of duodecimal fractions, the contributions to arithmetic.
unknown
to them.
numerical calculations were long and tedious, for the reason that they never possessed the boon of a perfect notation of
numbers with
its
zero and principle of local value.
GEOMETRY AND TRIGONOMETRY
EGYPT A3TD BABYLONIA crude beginnings of empirical geometry, like the art of that our counting, must be of very ancient origin. We suspect
THE
earliest records, reaching
back to about 2500
comparatively modern thought.
B.C.,
represent the
The Atones papyrus and
of geo Egyptian pyramids are probably the oldest evidences more it find "We convenient, however, to metrical study. is closely knitted with science Ancient with Babylonia.
begin
superstition.
We
have proofs that in Babylonia geometrical
1 were used in augury. Among these figures are a pair of parallel lines, a square, a figure with a reentrant angle, and
figures
an incomplete figure, believed to represent three concentric tri The accompany angles with their sides respectively parallel. word Sumerian the "line/' text contains meaning tim, ing "
"
hence the conjecture that the Babylonians, like the Egyptians, used ropes in measuring distances, and in is be determining certain angles. The Babylonian sign circle into six the lieved to be associated with the division of originally
rope
;
divided the circle into equal parts, and (as the Babylonians 360 degrees) with t^j)rigi^ofjhe^sexagesimal system. That this division into six parts (probably by the sixfold application of the radius) was known in Babylonia, follows from the in in the wheel of a royal carriage spection of the six spokes i
CAKTOE, Vol. I, pp. 4548. 43
A HISTORY OF MATHEMATICS
4
represented in a drawing found in the remains of Nineveh, Like the Hebrews (1 Kings vii. 23), the Babylonians took the ratio of the circumference to the diameter equal to 3, a de
Of geometrical demonstrations there mind the intuitive
cidedly inaccurate value. is
no
"As
trace.
a
rule, in the Oriental
powers eclipse the severely rational and logical." We begin our account of Egyptian geometry with the geo metrical problems of the Ahmes papyrus, which are found in the middle of the arithmetical matter.
Calculations of the
solid contents of barns precede the determination of areas. 1
Not knowing the shape
of the barns, we cannot verify the cor rectness of the computations, but in plane geometry Ahmes's
He considers the area of land in the forms of square, oblong, isosceles triangle, isosceles trapezoid, and circle. Example No. 44 gives 100 as the area of a square figures usually help us.
whose
sides are 10.
whose
sides are 10 ruths
In No. 51 he draws an isosceles triangle and whose base is 4 ruths, and finds
the area to be 20.
The
correct value
is
19.6".
Ahmes
J
s
obtained by taking the product of one approximation leg and half the base. The same error occurs in the area of the isosceles trapezoid. Half the sum of the two bases is is
multiplied by one leg. quadrature, for
it
His treatment of the
teaches
how
circle is
an actual
to find a square equivalent to
He takes as the side, the diameter dimin ished by of itself. This is a fair approximation, for, if the radius is taken as unity, then the side of the and square is the circular area.
^
its
area
(^)
2
==
3.1604
.
.
.
Besides these problems there are
others relating to pyramids and disclosing similar figures, of proportion, and,
some knowledge
of
perhaps, of rudimentary
2
trigonometry. Besides the
Ahmes
papyrus, proofs of the existence of
1
Gow,
2
Consult CANTOR, Vol. I, pp. 99102
pp. 126130. ;
Gow,
p. 128.
EGYPT AND BABYLONIA
45
ancient Egyptian geometry are found in figures on the walls The wall was ruled with squares, or other
of old structures.
rectilinear figures, within
which coloured pictures were drawn. (about 460370 B.C.) is 1
The Greek philosopher Democritus
" quoted as saying that in the construction of plane figures no one has yet surpassed me, not even the socalled HarpedoCantor has pointed out the meaning of the naptoe, of Egypt." .
word ;; harpedonaptse "
to be " ropestretchers."
.
.
2
This, together
with other clues, led him to the conclusion that in laying out temples the Egyptians determined by accurate astronomical observation a north and south line line at right angles to this
by means
;
then they constructed a around
of a rope stretched
three pegs in such a way that the three sides of the triangle formed are to each other as 3 4 5, and that one of the legs of :
:
this right triangle coincided with the
the other leg gave the E. and of the temple.
W.
N. and
S. line.
Then
line for the exact orientation
According to a leathern document in the Ber
" ropestretching occurred at the very early age of Amenemhat I. If Cantor's explanation is correct, it fol
lin
Museum,
"
lows, therefore, that the Egyptians were familiar with the
wellknown property of the righttriangle, in case of sides in the ratio 3:4:5, as early as 2000 B.C. Prom what has been said it follows that Egyptian geometry
What
flourished at a very early age.
subsequent centuries ?
In 257
B.C.
about
progress in
its
laid out the temple of B.C.
the number
owned by the priesthood, and
their areas,
Horus, at Edfu, in Upper Egypt. of pieces of land
was
About 100
were inscribed upon the walls in hieroglyphics. The incorrect formula of Ahmes for the isosceles trapezoid is here applied
any trapezium, however more than 2000 B.C. yield for
irregular.
1
Consiilt the drawings reproduced in
3
Ibidem,
p. 104.
Thus the formulae
of
closer approximations than those
CANTOR, Yol.
I.
,
p. 108.
A HISTORY OF MATHEMATICS
46
written two centuries after Euclid!
The
conclusion irresist
ibly follows that the Egyptians resembled the Chinese in the stationary character, not only of their government, but also of
An
their science.
explanation of this has been sought in the
fact that their discoveries in mathematics, as also in medicine, were entered at an early time upon their sacred books, and that, in after ages, it
augment anything
was considered heretical to modify or Thus the books themselves closed
therein.
the gates to progress.
Egyptian geometry is mainly, though not entirely, a geome try of areas, for the measurement of figures and of solids con
main part of it. This practical geometry can be called a science. In vain we look for theorems and hardly proofs, or for a logical system based on axioms and postulates. It is evident that some of the rules were purely empirical. stitutes the
If
we may
trust the testimony of the Greeks,
geometry had its origin in the surveying by the frequent overflow of the Kile.
Egyptian
of land necessitated
GBEECE About the seventh century B.C. there arose between Greece and Egypt a lively intellectual as well as commercial inter course. Just as Americans in our time go to Germany to so study, early Greek scholars visited the land of the pyra mids.
Thales,
Eudoxus, struction.
all
(Enopides,
Pythagoras,
sat at the feet of the
While Greek culture
commands our mind of the Greek
it
merely to the
is,
Plato,
Democritus,
Egyptian priests for in therefore, not primitive,
The speculative at once transcended questions pertaining practical wants of everyday life; it pierced enthusiastic
admiration.
into the ideal relations of things, and revelled in the study

A 47
GREECE of science as science.
always
be
admired,
For
this reason
notwithstanding
Greek geometry will its limitations and
defects.
Eudemus, a pupil of Aristotle, wrote a history of geometry. This history has been lost but an abstract of it, made by Proclus in Ms commentaries on Euclid, is extant, and is the most ;
trustworthy
information
We
geometry.
we have regarding
shall quote the account
early
Greek
under the name of
Eudem ia n Summary. The study of geometry was introduced by Thales of !Miletus (640546 B.C.), one of the u seven wise men." Commercial pursuits brought him to Egypt; intellectual pursuits, for a time, detained him there. Plutarch declares that Thales soon excelled the priests, and amazed King Amasis by measuring the heights of the pyra mids from their shadows. According to Plutarch this was I.
The Ionic School.
into Greece
done thus The length of the shadow of the pyramid is to the shadow of a vertical staff, as the unknown height of the pyra mid is to the known length of the staff. But according to :
Diogenes Laertius the measurement was different The height pyramid was taken equal to the length of its shadow at :
of the
the its
moment when the shadow own length, 1
of a vertical staff
was equal
to
The first method implies a knowledge of the proportionality of the sides of equiangular triangles, which some critics are unwilling to grant Thales. The rudiments of proportion were certainly known to Ahmes 1
and the builders of the pyramids. Allman grants Thales this knowledge, and, in general, assigns to him and his school a high rank. See Greek Geometry from Thales to Euclid, hy GEORGE JOHNSTON ALLMAN, Dublin,
Gow (p. 142) also believes in Plutarch's narrative, but 135) leaves the question open, while Hankel (p. 90), Bretschneider, Tannery, Loria, are inclined to deny Thales a knowledge of similitude of figures. See Die Geometrie und die Geometer vor Euklides. 1889, p. 14.
Cantor
Ein
La
(I., p.
Historischer Versuch, von C. A. BEETSCHXEIBER, Leipzig, 1870, p. 46 Geometrie Grecque . . . par PAUL TAKKEBY, Paris, 1887, p. 92 ; Le ;
A HISTOBY OF MATHEMATICS
48
The Eudernian Summary
ascribes to Thales the invention of
the theorems on the equality of vertical angles, the equality of the base angles in isosceles triangles, the bisection of the
by any diameter, and the congruence of two triangles having a side and the two adjacent angles equal respectively. circle
Famous
his application of the last theorem to the deter
is
the distances of ships from the shore. The all angles inscribed in a semicircle are right attributed by some ancients to Thales, by others to
mination of
theorem that angles
is
Thales thus seems to have originated the geome
Pythagoras. try of
lines
and
of angles, essentially abstract in character,
while the Egyptians dealt primarily with the geometry of sur faces
and of
solids, empirical in character.
1
It
would seem
as
though the Egyptian priests who cultivated geometry ought at least to have felt the truth of the above theorems. We have
no doubt that they
did, but
we
incline to the opinion that
Thales, like a true philosopher, formulated into theorems
subjected to proof that
which others merely
If this view is correct, then
felt to
and
be true.
follows that Thales in his pyra mid and ship measurements was the first to apply theoretical geometry to practical uses. it
Thales acquired great celebrity by the prediction of a solar With him begins the study of scientific eclipse in 585 B.C. astronomy. The story goes that once, while viewing the stars, he fell into a ditch. An old woman attending him exclaimed,
"How
canst thou
know what
thou seest not what
is
doing in the heavens, " at thy feet ? is
when
Astronomers of the Ionic school are Anaximander and Anaximenes.
Anaocagoras, a pupil of the latter, attempted to square the circle while he was confined in prison. Approxi* Scienze Esatte nelV Antica Gretia, di GINO LOKIA, in Modena, 1893,
Libro L, p. 25. 1
ALLMAN,
p. 16.
49
GREECE
mations to the ratio K occur early among the Egyptians, Baby the first recorded as lonians, and Hebrews but Anaxagoras is ;
that knotty ratio attempting the determination of the exact his time has been unsuccessfully attacked since which problem
no by thousands. Anaxagoras apparently offered The life of School. II. TJie Pythagorean
solution.
Pythagoras
We
enveloped in a deep mythical haze. (5S07500? are reasonably certain, however, that he was born in Samos, studied in and, later, returned to the place of his B.C.) is
Egypt, Perhaps he visited Babylon. .Failing in his attempt to found a school in Samos, be followed the current of civiliza tion, and settled at Croton in South Italy (Magna Graecia). There he founded the Pythagorean brotherhood with observ nativity.
Its members were ances approaching Masonic peculiarity. forbidden to divulge the discoveries and doctrines of their Hence it is now impossible to tell to whom individu school.
ally the various
Pythagorean discoveries must be ascribed.
was the custom among the Pythagoreans
It
to refer every dis
At first the school covery to the great founder of the sect. on account of an became it later but suspicion object flourished, of its mystic observances.
A
political party in
destroyed the buildings; Pythagoras
Metapontum.
Though
politics
fled,
Lower
Italy
but was killed at
broke up the Pythagorean fra
for at least two centuries. ternity, the school continued to exist Like Thales, Pythagoras wrote no mathematical treatises.
Pythagoras changed the study says of geometry into the form of a liberal education, for he exam ined its principles to the bottom, and investigated its theorems
The Eudemian Summary
:
in an immaterial and intellectual manner."
To Pythagoras
himself
is
to be ascribed the wellknown
truth of the theorem for property of the right triangle. The the special case when the sides are 3, 4, and 5, respectively, he are told that Pyhave learned from the Egyptians.
may
We
A HISTORY OF MATHEMATICS
50 fchagoras
is
was
so jubilant over this great discovery that
he sac
muses who inspired him. That this but a legend seems plain from the fact that the Pythagoreans
rificed a
hecatomb
to the
believed in the transmigration of the soul, and, for that reason, In the traditions of the late opposed the shedding of blood. ]STeoPythagoreans the objection is removed by replacing the The dem sacrifice that of " an ox made of flour "
by law
bloody
onstration of the
!
of three squares, given in Euclid,
has not been handed
I.,
47,
The proof given by jpythagoras
due to Euclid himself.
is
down
IffiSETingenuity has been expended in conjectures as to its nature. Bretschneider's sur mise, that the Pythagorean proof was substantially the same to "as:
as the one of Bhaskara (given elsewhere), has been well received by Hankel, Allman, Grow, Loria. 1 Cantor thinks it not improbable that the early proof involved the consideration of special cases, of which the isosceles
right triangle, perhaps the first, may have been proved in the manner indi
by the
cated
The
2
adjoining
figure.
four lower triangles together equal the four upper. Divisions of squares by their
diagonals in this fashion occur in Plato's Since the time of the Greeks the famous Pythagorean
Meno.
Theorem has received many
A
school 1
2
jwas
and his an
the combination of geometry and arithmetic;
See BRETSCHNEIDER, p. 82 LORIA, I., p. 48.
p. 155
different demonstrations. 3
characteristic point in the method of Pythagoras
;
ALLMAN,
p.
36
;
HANKEL,
p.
98
;
Gow,
;
CANTOR,
I.,
185; see also
Gow,
p. 155,
and ALLMAN,
p. 29.
3
See JOH. Jos. JGN. HOFFMANN, Der Pythagorische Lehrsatz mit zwey und dreysig theils bekannten^ theils neuen JBeweisen. Mainz, 1819. See also JURY WIPPER, Sechsundvierzig Beweise des Pythagoraischen Lehrsatzes. Aus dem Russischen von F. GRAAP, Leipzig, 1880, The largest collection of proofs is by F. Yanney and J. Math. Monthly, Vols. 3 and 4, 1896 and 1897.
A. Calderhead in the Am.
GREECE
51
arithmetical fact has its analogue in geometry, and vice versa. Thus in connection with the law of three squares Pythagoras
devised a rule for finding integral numbers representing the l as one lengths of the sides of right triangles: Choose 2?i
+
side,
then i[(2?i
f
If 1] = 2 H
2
f 2?i==
the other side, and 5, then the three
If n = hypotenuse. sides are 11, 60, 61. This rule yields only triangles whose hypotenuse exceeds one of the sides by unity.
2 n2
+ 2 n f 1 = the
^
Ascribed to the Pythagoreans
is
matical discoveries of antiquity
The discovery
is
t^JJ^
one of the "greatest mathe
that of Irrational Quantities.
usually supposed to have gro\vn out of the If each of the equal
1 study of the isosceles right triangle.
legs is taken as unity,
then the hypotenuse, being equal cannot be V2, exactly represented by any number what ever. TTe may imagine that other numbers, say 7 or i, were taken to represent the legs in these and all other cases experi to
;
mented upon, no number could be found
to exactly
measure the
After repeated failures, doubtless, length of the hypotenuse. " some rare genius, to whom it is granted, during some happy moments, to soar with eagle's flight above the level of human
may have been Pythagoras himself,
grasped the 2 that solved." As there cannot be this problem happy thought which could in the of was nothing shape any geometrical figure thinking,
it
suggest to the eye the existence of irrationals, their discovery The must have resulted from unaided abstract thought.
Pythagoreans saw in irrationals a symbol of the unspeakable. The one who first divulged their theory is said to have suf fered shipwreck in consequence, "for the unspeakable and invisible should always be kept
secrt^if___

1 ALLMJLS, p. 42, thinks it more likely that the discovery to cut a line in extreme and mean ratio. the problem
2 3
EANXEL, p. 101. The same story of death
in the sea
is
~~
""
was owing to
told of the Pythagorean Hip*
pasus for divulging the knowledge of the dodecaedron.
A HISTOBY OF MATHEMATICS
52
The theory
of parallel lines enters into the
proof of the anglesum in triangles; a line being
Pythagorean drawn parallel
In this mode of proof we observe progress from the special to the general, for according to Geminus the early demonstration (by Thales?) of this theorem embraced three to the base.
different cases: that of equilateral, that of isosceles, and, finally, that of scalene triangles. 1 Eudenius says the Pythagoreans invented the problems con cerning the application of areas, including the cases of defect
and
excess, as in Euclid, VI., 28, 29. They could also construct a polygon equal in area to a given polygon and similar to another. In a general way it may be said that the Pythago
rean plane geometry, like the Egyptian, was much concerned with areas. Conspicuous is the absence of theorems on the circle.^
^^4,
.
The Pythagoreans demonstrated a point
is
completely
filled
by
also that the plane about
six equilateral triangles, four
squares, or three regular hexagons, so that a plane can be divided into figures of either kind. Eelated to the study of regular polygons is that of the regular solids. It is here that the Pythagoreans contributed to solid geometry. From the equilateral triangle and the square arise the tetraedron, all four probably known to octaedron, cube, and icosaedron the Egyptians, certainly the first three. In Pythagorean phi losophy these solids represent, respectively, the four elements
of the physical world: fire, air, earth, and water. In absence of a fifth element, the subsequent discovery of the dodecaedron
was made to represent the universe itself. The legend goes that Hippasus perished at sea because he divulged "the sphere with the twelve pentagons." Pythagoras used to say that the most beautiful of
was the sphere
;
of all plane figures, the circle. 1
Consult HAKKEL, pp. 95, 96.
all solids
GREECE With what
53
degree of rigour the Italian school demonstrated
we have no sure way of determining. We are in however, assuming that the progress from empirical to safe, reasoned solutions was slow.
their theorems
Passing to the later Pythagoreans, we meet the name of Philolaus, who wrote a book on Pythagorean doctrines, which he made known to the world. Lastly, we mention the brilliant
Tarentum (428347 B.C.), who was the only great geometer in Greece when Plato opened his school. He ad vanced the theory of proportion and wrote on the duplication Arcliytas of
of the cube. III.
1
The Sophist School
The
periods of existence of the
several Greek mathematical "schools" overlap considerably. Thus, Pythagorean activity continued during the time of the sophists, until the opening of the Platonic school.
After the repulse of the Persians at the battle of Salamis and the expulsion of the Phoenicians and pirates
in 480 B.C.
from the ^Egean Sea, Greek commerce began to flourish. Athens gained great ascendancy, and became the centre toward which Pythagoreans flocked thither; Anaxag
scholars gravitated.
oras brought to Athens Ionic philosophy. The Pythagorean to the of ceased be observed; secrecy spirit of Athe practice 2
All menial work being per publicity. the Athenians were formed by slaves, people of leisure. That they might excel in public discussions on philosophic or scien
nian
tific
life
demanded
questions, they
for teachers,
must be educated.
who were
There arose a " called sophists, or wise men."
demand Unlike
the early Pythagoreans, the sophists accepted pay for their teaching. They taught principally rhetoric, but also philoso
phy, mathematics, and astronomy.
The geometry was now taken 1
of the circle, neglected up.
The researches
Consult ALLMAN, pp. 102127.
by the Pythagoreans, of the sophists centre 2
ALLMAN,
p. 54.
A HISTOBT OF MATHEMATICS
54
around the three following famous problems, which were to be constructed with aid only of a ruler and a pair of compasses :
(1) (2)
The The
whose
trisection of
any
angle or arc
duplication of the cube
;
;
i.e.
to construct a cube
volume shall be double that of a given cube;
(3) The squaring of the circle ; i.e. to construct a square or other rectilinear figure whose area exactly equals the area of a
given
circle.
Certainly no other problems in mathematics have been stud and persistently as these. The best Greek
ied so assiduously
was bent upon them Arabic learning was applied to of the best mathematicians of the Western Renais some them; sance wrestled with them. Trained minds and untrained minds, wise men and cranks, all endeavoured to conquer these prob lems which the best brains of preceding ages had tried but failed to solve. At last the fact dawned upon the minds of men that, intellect
;
so long as they limited themselves to the postulates laid
down
by the Greeks, these problems did not admit of solution. This divination was later confirmed by rigorous proof. The Greeks demanded constructions of these problems by ruler and com passes, but no other instruments.
was
In other words, the figure and of circles. A con
to consist only of straight lines
struction
was not geometrical
if
effected
by drawing
parabolas, hyperbolas, or other higher curves.
such curves the Greeks themselves resolved
all
ellipses,
With
aid of
three problems;
but such solutions were objected to as mechanical, whereby " the good of geometry is set aside and destroyed, for we again reduce it to the world of sense, instead of elevating and imbu ing
it
with the eternal and incorporeal images of thought, even " employed by God, for which reason He always is God
as it is
(Plato).
Why should
the Greeks have admitted the circle into
geometrical constructions, but rejected the ellipse, parabola, curves of the same order as the circle?
and the hyperbola
GREECE
We
answer in the words of
55
Sir Isaac
Xewton
l :
" It
is
not the
simplicity of the equation, but the easiness of the description, which is to determine the choice of our lines for the construc .
For the equation that expresses a parabola is more simple than that that expresses a circle, and yet the 772 it. circle, by its more simple construction, is admitted before tions of problems.
The
bisection of an angle
one of the easiest of geometrical Early investigators, as also beginners in our elementary classes, doubtless expected the division of an angle In the special case into three equal parts to be fully as easy. is
constructions.
of a right angle the construction
is
readily found, but the
One Hippias, very general case offers insuperable probably Hippias of Elis (born about 460 B.C.), was among difficulties.
the earliest to study this problem. tion involving merely circles
a transcendental curve
(i.e.
Failing to find a construc
and straight lines, he discovered one which cannot be represented
by an algebraic equation) by which an angle could be divided not only into three, but into any number of equal parts. As this same curve was used later in the quadrature of the received the
circle, it
name
of quadratrix*
1 ISAAC NEWTON, Universal Arithmetick. Translated by the late Mr. Ralphson revised by Mr. Cunn. London, 1769, p. 468. 3 In one of De Morgan's letters to Sir W. R. Hamilton occurs the fol tk But what distinguishes the straight line and circle more than lowing: ;
anything else, and properly separates them for the purpose of elementary geometry ? Their self similarity. Every inch off a straight line coincides with every other inch, and off a circle with every other off the same circle. Where, then, did Euclid fail ? In not introducing the third curve,
which has the same property
the screw.
The
right line, the circle, the
the representatives of translation, rotation, and the two combined we ought to have been the instruments of geometry. With a screw
screw
should never have heard of the impossibility of trisecting an angle, squar GRAVES, Life of Sir William Rowan Hamilton, ing the circle," etc. of easiness of descrip 1889, Vol. III., p. 343. However, if Newton's test tion be applied, then the screw must be excluded. 8 For a That the indescription of the quadratrix, see Gow, p. 164.
A HISTORY OF MATHEMATICS
6
The problem to geometers as
lem
" to double the cube " perhaps suggested itself
an extension to three dimensions of the prob
upon the diag area of this the constructed,
in plane geometry, to double a square.
onal of a square a
new square
is
If
new is
square is exactly twice the area of the first square. This But the at once evident from the Pythagorean Theorem.
construction of a cube double in volume to a given cube brought to light unlookedfor difficulties. to this
A.
different origin is assigned
The Delians were once
problem by Eratosthenes.
suf
fering from a pestilence, and were ordered by the oracle
double a certain cubical
to
Thoughtless workmen simply
altar.
constructed a cube with edges twice as long; but brainless work like that did not pacify the gods. The error being dis covered, Plato
was consulted on
this
" Delian problem."
Era
tosthenes tells us a second story King Minos is represented by an old tragic poet as wishing to erect a tomb for his son :
;
being dissatisfied with the dimensions proposed by the archi tect, the king exclaimed: "Double it, but fail not in the cubical form."
originated in
(about 430
we
If
trust these stories, then the problem
an architectural
B.C.)
was the
1
difficulty.
show that
first to
Hippocrates of Chios this problem can be
reduced to that of finding between a given line and another twice as long, two mean proportionals, i.e. of, inserting two lengths between the lines, so that the four shall be in geomet
In modern notation, if a and 2 a are the and x and y the mean proportionals, we have the
rical progression.
two
lines,
progression 2
y
=2ax.
a, x, y,
Then
a;
2 a, which gives 4
=
a~f
= 2 a x, z
 = = x y?
^ ;
whence x2 =ay,
2a y = 2 a3 But Hippocrates .
ventor of the quadratrix was Hippias of Ells is denied by HANKEL, p. 151, and ALLMAN, p. 94 but affirmed by CANTOR, I., p. 181 ; BRETSCHNEIDEB, ;
p.
94 1
Gow, p. 163 Gow, p. 162. ;
;
LORIA,
I., p.
66
;
TANNEST, pp.
108, 131.
GREECE
57
naturally failed to find x9 the side of the double cube, by geo metrical construction. However, the reduction of the problem in solid geometry to one of plane geometry was in itself no mean achievement. He became celebrated also for his success in squaring a lune. This result he attempted to apply to the squaring of the circle. 1 In his study of the Delian and the quadrature problems Hippocrates contributed much to the geometry of the circle. The subject of similar figures, involv
ing the theory of proportion, also engaged his attention. He wrote a geometrical textbook, called the Elements (now lost),
whereby, no doubt, he contributed vastly towards the progress of geometry
by making
it
more
easily accessible to students.
said to have
once lost all his property. accounts say he fell into the hands of pirates, others attribute the loss to his own want of tact. Says Aristotle: " It is well known that in one persons stupid respect are by no means so in others. There is nothing strange in this so
Hippocrates
is
Some
:
skilled in geometry, appears to
have been
Hippocrates, though in other respects weak and stupid; and he lost, as they say, through his simplicity, a large sum of money by the fraud of the collectors of customs at Byzantium." 2
A considerable
step in advance was the introduction of the exhaustion by the sophist Antiphon, a contemporary process of of Hippocrates. By inscribing in a circle a square and on its sides erecting isosceles triangles with their vertices in the cir
cumference, and on the sides of these triangles erecting new he obtained a succession of regular in
isosceles triangles, etc.,
scribed polygons of 8, 16, 32, sides, and so on, each polygon approaching in area closer to that of the circle than did the
preceding polygon, until the circle was finally exhausted. 1 2
For
details see
Gow,
Anti
pp. 165168.
Translated by ALLMAN, p. 1247 a, 15, ed. Bekker.
p. 57,
from Arist. EtU. ad
J8ud. 9
VIL,
c.
XIY.,
A HISTORY OF MATHEMATICS
58
phon concluded that a polygon could be thus
inscribed, the
sides of which, on account of their minuteness,
would coincide
with the circumference of the circle. Since squares can be found exactly equal in area to any given polygon, there can be constructed a square exactly equal in area to the last poly gon inscribed, and therefore equal to the circle itself. Thus the possi appears that he claimed to have established of One his con circle. the of bility of the exact quadrature it
temporaries,
Bryson of Heradea, modified
exhaustion by
not only inscribing,
this
but also,
time, circumscribing regular polygons.
He
secure coincidence between the polygons
process at the
of
same
did not claim to
and
circle,
but he
committed a gross error by assuming the area of the circle be the exact arithmetical mean of the two polygonal areas.
to
Antiphon's attempted quadrature involved a point eagerly by philosophers of that time. All other Greek
discussed
geometers, so far as
we know, denied
coincidence of a polygon and a
the possibility of the
circle, for a straight line can
never coincide with a circumference or part of it. If a poly gon could coincide with a circle, then, says Simplicius, we would have to put aside the notion that magnitudes are divisi ble
ad
infinitum.
We
have here a
which
difficult
philosophical ques
Athens appears to have modified mathematical thought and Greek influenced greatly The Eleatic philosophical school, with in respect to method.
tion, the discussion of
at
the great dialectician Zeno at its head, argued with admirable ingenuity against the infinite divisibility of a line, or other
Zeno's position was practically taken by Antiphon in his assumption that straight and curved lines are ultimately
magnitude.
1 reducible to the same indivisible elements.
reductio
Zeno reasoned by
ad dbmrdum against the theory of the 1
ALLMAN,
p. 66,
infinite divisi
GREECE
59
He argued that if this theory is assumed to bility of a line. be correct, Achilles could not catch a tortoise. For while he ran to the place where the tortoise had been when he started, the tortoise crept some distance ahead, and while Achilles hastened to that second spot, it again moved forward a little,
and
so on.
infinitely
Being thus obliged first to reach every one of the places which the tortoise had previously occu
many
pied, Achilles could never overtake the tortoise.
matter of
fact, Achilles
indefinite
number
But, as a
could catch a tortoise, therefore, it is to that the assume distance can be divided into an wrong
is
always at rest
of parts.
In
like
for it is at each
;
These paradoxes involving the
manner,
" the flying arrow
moment only
in one place."
infinite divisibility,
and there
fore the infinite multiplicity of parts no doubt greatly per plexed the mathematicians of the time. Desirous of construct
ing an unassailable geometric structure, they banished from their science the ideas of the infinitely little and the infinitely to
^Moreover,
great.
theorems evident secting
circles
meet other objections of
dialecticians,
to the senses (for instance, that
cannot
jected to rigorous
have
a
common
demonstration.
centre)
Thus the
two
inter
were sub
influence
of
dialecticians like Zeno, themselves not mathematicians, greatly modified geometric science in the direction of increased rigour. 1
The process of exhaustion, adopted by Antiphon and Bryson, was developed into the perfectly rigorous method of exhaustion. In finding, for example, the ratio between the areas of two circles, similar
number
polygons were inscribed and by increasing the
of sides, the spaces between the polygons
and
circles
nearly exhausted. Since the polygonal areas were to each other as the squares of the diameters, geometers doubtless divined the theorem attributed to Hippocrates of Chios, that 1
Consult further, HANEJBL, ,
I., p. 63.
p.
118
;
CANTOR,
I., p.
198
;
ALLMAN,
p. 55
;
A HISTORY OF MATHEMATICS
60
the circles themselves are to each, other as the square of their
But in order to exclude
diameters.
all
vagueness or possibility
of doubt, later Greek geometers applied reasoning like that in Euclid, XII., 2, which we give in condensed form as follows :
Let
(7,
c
be two circular areas
D
the proportion
if
D
2
cP
:
=C
:
c'.
If
c'
2 :
<
d2 c,
=C
;
D,
d,
the diameters.
Then,
not true, suppose that then a polygon p can be inscribed in :
c
is
the circle c which comes nearer to
c in
area than does
If
c'.
P be the corresponding polygon in then P:p=D :d =C: we have P> C, which and P:(7=p:c Since p > 2
2
(7,
f
.
absurd.
c',
Similarly,
nor smaller than
c,
c
it
f
cannot exceed
must be equal
As
c.
to
c.
c'
c',
is
cannot be larger,
Q.E.D.
Here we have
the process of exemplified the method of exhaustion, involving Hankel refers reasoning, designated the reductio ad absurdum. this
method of exhaustion back
the reasons for assigning to Eudoxus,
seem
it
to Hippocrates of Chios, but
to this early writer, rather than
insufficient^*
After the Peloponnesian War the political power of Athens declined, but her (431404 B.C.), leadership in philosophy, literature, and science became all the IV.
The Platonic School
She brought forth such men as Plato (429 ?347 B.C.), the strength of whose mind has influenced philosophical stronger.
Socrates, his early teacher, despised thought of all ages. mathematics. But after the death of Socrates, Plato travelled
extensively and came in contact with several prominent math At Gyrene he studied geometry with Theodorus ematicians. the Pythagoreans. Archytas of Tarentum and met he in Italy ;
Timseus of Locri became his intimate friends.
About 389
B.C.
Plato returned to Athens, founded a school in the groves of the Academia, and devoted the remainder of his life to teach ing and writing. i
Unlike his master, Socrates, Plato placed
Consult HJLNKEL, p. 122
j
Gow,
p. 173
;
CANTOR, L, pp.
242, 247.
GKEECE
61
great value upon the minddeveloping power of mathematics. " Let no one who is unacquainted with geometry enter here," was inscribed over the entrance to his school. Likewise Xenocrates, a successor of Plato, as teacher in the
Academy, declined
to admit a pupil without mathematical training, "Depart, for thou hast not the grip of philosophy." The Eudemian Sum
mary says of Plato that "he filled his writings with mathe matical terms and illustrations, and exhibited on every occa sion the remarkable
connection
between mathematics
and
philosophy." Plato was not a professed mathematician. He did little or no original work, but he encouraged mathematical study and
suggested improvements in the logic and methods employed in geometry. He turned the instinctive logic of the earlier geometers into a method to be used consciously and without 1
With him begin careful definitions and the misgiving. consideration of postulates and axioms. The Pythagorean definition,
"a point is unity in position,"
sophical theory, was
embodying a philo
rejected by the Platonists a point was defined as "the beginning of a straight line " or "an indivisible line."
;
According to Aristotle the following definitions were
also current
:
The
point, the line, the surface, are respectively
the boundaries of the line, the surface, and the solid a solid is that which has three dimensions. Aristotle quotes from ;
the Platonists the axiom
If equals be taken from equals, the remainders are equal. How many of the definitions and axi oms were due to Plato himself we cannot tell. Proclus and :
Diogenes Laertius name Plato as the inventor of the method of proof called analysis. To be sure, this method had been
used unconsciously by Hippocrates and others, but it is gen erally believed that Plato was the one who turned the un1
Grow? pp. 175, 176.
See also HANKEL, pp. 127150.
A HISTORY OF MATHEMATICS
62
conscious logic into a conscious,
development and perfection of great achievement, but ascribe
it
Allman
legitimate method.
this (p.
The method was certainly a 125) is more inclined to
Archytas than to Plato.
to
The terms
and analysis in Greek mathematics had what they have in modern mathe
synthesis
a different meaning from
matics or in logic. 1
The
oldest definition of analysis as op
posed to synthesis is given in Euclid, XIII., 5, likely framed by Eudoxus
2
"
which was most
the obtaining of Analysis the thing sought by assuming it and so reasoning up to an admitted truth synthesis is the obtaining of the thing sought is
:
;
by reasoning up
to the inference
and proof of
it."
3
Plato gave a powerful stimulus to the study of solid geom The sphere and the regular solids had been studied
etry.
The
somewhat by the Pythagoreans and Egyptians. 1
HANKEL,
pp. 137150
j
also
Jahrhunderten, Tubingen, 1884,
latter,
in den letzten
HANKEL, MatJiematik p. 12.
2
BEETSCHNEIDER, p. 168. Greek mathematics exhibits different types of analysis. One is the reductio ad absurdum, in the method of exhaustion. Suppose we wish to prove that "A is B." We assume that A is not B then we form a 8
;
synthetic series of conclusions : not .B is C, not E, then it is impossible that A is not ;
B
this process
by taking Euclid, XII.,
2,
C is i.e.,
D,
A
D is E is
given above.
B.
;
if
now
A
is
Q.E.D.
Verify Allied to this is the
To prove that A is B assume that A is B, then B is (7, C is Z>, D is E, E is F; hence A is F. If this last is known to be false, then A is not B if it is known to be true, then the reasoning thus far is not conclusive. To remove doubt we must follow the reverse process, theoretic analysis
A
F
:
E
D
A
is F, is E, is D, is C, C is JB; therefore is B. This second case involves two processes, the analytic followed by the synthetic. The only aim of the analytic is to aid in the discovery of the synthetic. Of
greater importance to the Greeks was the problematic analysis, applied in constructions intended to satisfy given conditions. The construction
assumed as accomplished then the geometric relations are studied with the view of discovering a synthetic solution of the problem. For examples of proofs by analysis, consult HAXKEL, p. 143 Gow, p. 178 ; ALLMAX, pp. 160163 TODHDNTEB'S Euclid, 1869, Appendix, pp. 320328. is
;
;
;
GREECE
63
more or less familiar with, the geometry of In the Platonic school, the prism, pyramid, The study of the cone cylinder, and cone were investigated. to the led Mensechmus discovery of the conic sections. Per of course, were
the pyramid.
haps the most brilliant mathematician of this period was He was born at Cnidus about 408 B.C., studied
Eudoxm.
under Archytas and, for two months, under Plato. Later he taught at Cyzicus. At one time he visited, with his pupils,
He died at Cyzicus in 355 B.C. Among the pupils of Eudosus at Cyzicus who afterwards entered the academy of Plato were Meneechmus, Dinostratus, Athenaeus, the Platonic school.
and Helicon.
The fame of the Academy is largely due to The Eudemian Summary says that Eudoxus "first increased the number of general theorems, added to the three
these.
proportions three more, and raised to a considerable quantity the learning, begun by Plato, on the subject of the section, to which he applied the analytic method." By this " section " " is meant, no doubt, the golden section," which cuts a line
and mean ratio. He proved, says Archimedes, that a pyramid is exactly onethird of a prism, and a cone That onethird of a cylinder, having equal base and altitude.
in extreme
spheres are to each other as the cubes of their radii, was The method of exhaustion probably established by him.
was used by him extensively and was probably
his
own
in
vention. 1 1
The Eudemian Summary mentions,
named, Thesetetus
of Athens, to
whom
besides the geometers already is supposed to he indebted
Euclid
in the composition of the 10th book, treating of incommensurables (ALLMAN, pp. 206215); Leodamas of Thasos ; Neocleides and his pupil Leon,
who wrote a geometry
;
Theudius
of
Magnesia,
who
also
wrote a geometry
or Elements ; Hermotimus of Colophon, who discovered many propositions in Euclid's Elements; Amyclas of Heraclea, Cyzicenus of Athens, and
Philippus of Mende. not extant.
The preEuclidean textbooks
just
mentioned are
A HISTORY OF MATHEMATICS
64
During the
The First Alexandrian School.
V.
sixtysix
a period of political years following the Peloponnesian War Athens brought forth some of the greatest and most decline
In 338 B.C. she was con Greek antiquity. and her of Macedon power broken forever. quered by Philip Soon after, Alexandria was founded by Alexander the Great,
subtle thinkers of
and
it
and
art
was
in this
city that literature, philosophy,
science,
found a new home.
In the course of our narrative, we have seen geometry take feeble root in Egypt; we have seen it transplanted to the Ionian Isles thence to Lower Italy and to Athens now, at ;
;
grown
last,
transferred to ated,
and graceful proportions, we see it the land of its origin, and there, newly invigor
to substantial
expand in exuberant growth.
Perhaps the founder, certainly a central figure, of the Alex andrian mathematical school was Euclid (about 300 B.C. ). 2s~o ancient writer in any branch of knowledge has held such a com
modern education as has Euclid in ele " The sacred writings excepted, no Greek mentary geometry. l has been so much read or so variously translated as Euclid." manding
position in
After mentioning Eudoxns, Thesetetus, and other members 2 adds the following to the En
of the Platonic school, Proclus
demian Summary:
"Not much ments,
later than these is Euclid,
arranged much
who wrote the Ele much
of Eudoxns's work, completed
of Thesetetus's, and brought to irrefragable proof propositions which had been less strictly proved by his predecessors.
Euclid lived during the reign of the
first
Ptolemy, for he
is
quoted by Archimedes in his first book and it is said, more over, that Ptolemy once asked him, whether in geometric mat;
1
DE MOKGAN, " Eucleides " in Smith's Dictionary of Greek and fioman
Biography and Mythology. 1
We commend
PKOCLUS (Ed. ITBIEDLEIN),
p. 68.
this
remarkable
article to alL
GREECE ters there
65
was not a shorter path than through
his Elements,
1 to which he replied that there was no royal road to geometry. He is, therefore, younger than the pupils of Plato, but older
than Eratosthenes and Archimedes, for these are contempora ries,
as Eratosthenes informs us.
He belonged
to the Platonic
and was familiar with Platonic philosophy, so much so, in fact, that he set forth the final aim of his work on the Elements
sect
to be the construction of the socalled Platonic figures (regular 2
3 Pleasing are the remarks of Pappus, who says he who could in the least was gentle and amiable to all those
solids)."
degree advance mathematical science.
lowing story Euclid,
" :
when he had
What do I get by his slave and said, '
Stobaeus* tells the fol
A youth who had begun to learned the
read geometry with
first
proposition, inquired, 7 So Euclid called learning those things ? '
Give him threepence, since he must gain
" out of what he learns.' is given in these extracts do we life All other statements of Euclid. the concerning about him are trivial, of doubtful authority, or clearly
Kot much more than what
know
1 "This piece of wit has had many imitators. 'Quel diable,' said a French nobleman to Rohault, his teacher in geometry, pourrait entendre Ce serait un diable qui aurait de la cela ? to which the answer was, A story similar to that of Euclid is related by Seneca (Ep. 91, patience.' cited by August) of Alexander." DE MORGAN, op. cit. 2 This statement of Euclid's aim is obviously erroneous. 8 PAPPUS (Ed. HULTSCH), pp. 676678. * Quoted by Gow, p. 195 from Floril. IV M p. 250. 5 Syrian and Arabian writers claim to possess more information about Euclid they say that his father was Naucrates, that Euclid was a Greek born in Tyre, that he lived in Damascus and edited the Elements of ApolFor extracts from Arabic authors and for a list of books on the lonius. '
*
'
;
18. During principal editions of Euclid, see LORIA, II., pp. 10, 11, 17, the middle ages the geometer Euclid was confused with Euclid of Megara, a pupil of Socrates. Of interest is the following quotation from DE MOB*
A HISTORY OF MATHEMATICS
66
Though the author
of
several
works on mathematics and
physics, the fame of Euclid has at all times rested mainly upon This book was so his book on geometry, called the Elements.
by Hippocrates, Leon, and the latter works soon TheudiuSj that perished in the struggle The great role that it has played in geometric for existence. far superior to the Elements written
all subsequent centuries, as also its strong and viewed in the light of pedagogical science and of points, modern geometrical discoveries, will be discussed more fully
teaching during
weak
later.
At present we
account of
its
confine ourselves to a brief critical
contents.
Exactly how much of the Elements is original with Euclid, we have no means of ascertaining. Positive we are that certain early editors of the Ele?nents were wrong in their view that a finished and unassailable system of geometry sprang at once from the brain of Euclid, " an armed Minerva from the head of Jupiter." Historical research has shown that Euclid got
the larger part of his material from the eminent mathema " Theorem ticians who preceded him. In fact, the proof of the " to him. one ascribed of Pythagoras is the only directly
Allman 1 conjectures that the substance
of Books
I.,
II.,
IV.
comes from the Pythagoreans, that the substance of Book YI. is due to the Pythagoreans and Eudoxus, the latter contribut ing the doctrine of proportion as applicable to incommensurables and also the Method of Exhaustions (Book XII.), that Theaetetus contributed
much toward Books X. and XTIL,
that
work of Euclid himself
is to
the principal part of the original
" In the GAN, op. cit.; frontispiece to Winston's translation of Tacquet's Euclid there is a bust, which is said to be taken from a brass coin in pos
but no such coin appears in the published queen of Sweden. Sidonius Apollinaris says (Epist. XL, 9) that it was the custom to paint Euclid with the fingers extended (laxatis), as if in the act of measurement." session of Christina of
Sweden
;
collection of those in the cabinet of the
1
ALLMAN,
pp. 211, 212.
GREECE oe found in
Book X.
The
67
greatest achievement of Euclid, no
doubt, was the coordinating and systematizing of the material handed down to him. He deserves to be ranked as one of the greatest sjstematizers of all time.
The
contents of the Elements
follows
Book
:
Books L,
II., III.,
may
be briefly indicated as
IV., VI. treat of plane
V., of the theory of proportion applicable to
geometry
;
magnitudes
in general; Books VII, VIII, IX. 3 of arithmetic; Book X., of the arithmetical characteristics of divisions of straight lines (i.e.
of irrationals)
XIII., XIV.,
XV,
are apocryphal,
;
Books XI., XII., of
solid
of the regular solids.
and are supposed
to
Hypsicles and Damascius, respectively.
Books two books
geometry
The
last
;
have been written by 1
Difference of opinion long existed regarding the merits of the scientific treatise. Some regarded it as a work
Elements as a
whose
logic is in every detail perfect and unassailable, while others pronounced it to be "riddled with fallacies." 2 In our opinion, neither view is correct. That the text of the Elements is
not free from faults
evident to any one who reads the com Perhaps no one ever surpassed Robert
is
mentators on Euclid.
Simson in admiration for the great Alexandrian. Yet Simson's "notes" to the text disclose numerous defects. While Simson
is certainly wrong in attributing to blundering editors the defects which he noticed in the Elements, early editors are doubtless responsible for many of them. Most of the
all
emendations pertain to points of minor importance. The work as a whole possesses a high standard of accuracy. A
minute examination of the text has disclosed to commentators 1 See Gow, p. 272 ; CANTOR, L, pp. 358, 501, Book XV., in the opinion of Heiberg and others, consists of three parts, of which the third is perhaps due to Damascius. See LOEIA, II., pp. 8892. 2 See C. S. PEIECE in Nation, Vol. 54, 1892, pp. 116, 366, and in the
Monist, July, 1892, p. 539 pp. 9193,
j
G. B. HALSTED, Educational Beview> 8, 1894
A HISTORY OF MATHEMATICS
68
an occasional lack of extreme precision in the statement of ^hat is assumed without proof, for truths are treated as selfevident
which are not found in the
list of
1
postulates.
Again, Euclid sometimes assumes what might be proved; as when in the very definitions he asserts that the diameter of a 2
which might be readily proved from defines a plane angle as the " inclination of two straight lines to one another, which meet together, but are not in the same straight line," but leaves the idea of angle
circle bisects the figure,
the axioms.
He
magnitude somewhat indefinite by his failure to give a test for equality of two angles or to state what constitutes the sum or difference of
two angles. 3
Sometimes Euclid
fails to consider
or give all the special cases necessary for the full and complete 4 Such instances of defects which friendly proof of a theorem. critics
have found in the Elements show that Euclid
infallible. 1
5
But
in noticing these faults,
For instance, the intersection
and
of the circles in L, 1,
See also H. M. TAYLOR'S Euclid, 1893, p. vii. 2 DE MORGAN, article "Euclid of Alexandria,"
in the
not
is
we must not in
lose
I.,
22.
English Cyclo
paedia. 8 See SIMQX XEWCOMB, Elements of Geometry, 1884, Preface H. M. TAYLOR'S Euclid, p. 8 DE MORGAN, The Connexion of Number and ;
;
Magnitude, London, 1836, p. 85. * Consult TODHUNTER'S Euclid, notes on L, 35; SIMSON'S Euclid, notes on L, 7 III., 35.
III.,
21
;
XI., 21.
;
6 A proof which has been repeatedly attacked is that of I., 16, which "uses no premises not as true in the case of spherical as in that of plane triangles and yet the conclusion drawn from these premises is known to be false of spherical triangles." See Nation, Vol. 54, pp. 116, 366; ;
E. T.Dixox, in Association for the Improvement of Geometrical Teaching (A. I. G. T.), 17th General Report, 1891, p. 29 ; ENGEL und STACKEL, Die Theorie der Parallellinien von Euklid bis auf Gauss, Leipzig, 1895, (Hereafter we shall cite this book as ENGEL and STACKEL.) In the Monist, July, 1894, p. 485, G. B. Halsted defends the proof of I., 16, ** Two straight lines arguing that spherics are ruled out by the postulate, cannot enclose a space." If we were sure that Euclid used this postulate, then the proof of I., 16 would be unobjectionable, but it is probable that
p. 11, note.
GEEECB sight of the general excellence of the an excellence which in 1877
treatise
69
work as a scientific was fittingly recog
by a committee of the British Association for the Advancement of Science (comprising some of England's ablest mathematicians) in their report that "no textbook that has nized
yet been produced 7 1
authority/
is fit
to succeed Euclid in the position of
As already remarked, some editors of the Elements,
particularly [Robert Simson, wrote on the supposition that the
and any defects
in the text as
to them, they attributed to corruptions.
For example,
original Euclid
known
was
perfect,
Sim son thinks there should be a at the beginning of the fifth
book
definition of ;
compound
ratio
so he inserts one and assures
the very definition given by Euclid. Xot a single manuscript, however, supports him. The text of the Elements
us that
it is
now commonly used him the scapegoat
is
Theon's.
for all defects
Simson was inclined
to
make
which he thought he discov
But a copy of the Elements sent with other the Vatican to Paris by Napoleon I, is from manuscripts ered in Euclid.
believed to be anterior to Theon's recension; this differs but slightly
from Theon's version, showing that the
faults are
probably Euclid's own. At the beginning of our modern translations of the Elements (Bobert Simson's or Todhunter's, for example), under the head of definitions are given the assumptions of such notions as the Then follow point, line, etc., and some verbal explanations.
three postulates or demands, (1) that a line may be drawn from any point to any other, (2) that a line may be indefinitely pro duced, (3) that a circle may be drawn with any radius and any
point as centre.
After these come twelve axioms. 2
The term
the postulate was omitted by Euclid and supplied by some commentator. See HEIBEEG, Euclidis Elementa. 1
2
A. Of
I. G. T., 6th General Beport, 1878, p. 14. these twelve " axioms," five are supposed not to have been given
by
A HISTORY OF MATHEMATICS
70
axiom was used by Proclus, but not by Euclid.
He
speaks
common either to all men or to common notions " " axioms " relate to all kinds of sciences. The first nine mag
instead of " all
nitudes (things equal to the same thing are equal to each other, 1 etc., the whole is greater than its part), while the last three
(two straight lines cannot enclose a space all right angles are equal to one another the parallelaxiom) relate to space only. ;
;
While
in nearly all but the
most recent of modern editions
of Euclid the geometric "axioms" are placed in the same category with the other nine, it is certainly true that Euclid
sharply distinguished between the two classes. An immense " " preponderance of manuscripts places the axioms relating to 2
among the postulates. This is their proper place, for modern research has shown that they are assumptions and not common notions or axioms. It is not known who first made the unfortunate change. In this respect there should space
" two Euclid, viz. the four on inequalities and the one, straight lines can not enclose a space." Thus HEIBEEG and MENGE, in their Lathi and Greek edition of 1883, omit all five. See also ENGEL and STACKEL, p. 8, note. 1 "That the whole is greater than its part is not an axiom, as that emi Of finite collections it is true, nently bad reasoner, Euclid, made it to be of infinite collections false.'* C. S, PEIRCE, Monist, July, 1892, p. 539. Peirce gives illustrations in which, for infinite collections, the " axiom " is
Nevertheless, we are unwilling to admit that the assuming of axiom proves Euclid a bad reasoner. Euclid had nothing whatever to do with infinite collections. As for finite collections, it would seem that nothing can be more axiomatic. To infinite collections the terms great and small are inapplicable. On this point see GEOBG CANTOR,
untrue. this
"Ueber
eine Eigenschaft des Inbegriffes reeller algebraischer Zahlen," or for a more elementary discussion, see FELIX
Crelle Journal, 77, 1873
;
KLEIN, Ausgewahlte Fragen der Elementargeometrie, ausgearbeitet von F. TAGEET, Leipzig, 1895, p. 39. [This work will be cited hereafter as KLEIN.] That "the whole is greater than its part" does not apply in the comparison of infinites was recognized by Bolyai. See HALSTED'S BOLYAI'S Science Absolute of Space, 4th Ed., 1896, 24, p. 20. 
HANKEL, Die Complexen Zahlen
r
Leipzig, 1867, p. 52,
GREECE
<
The
oe a speedy return to Euclid's practice.
a very important
late plays
Most modern
1
parallelpostu
role in the history of geometry.
1
authorities hold that Euclid missed one of the
postulates, that of rigidity (or else that of equal variation), which demands that figures may be moved about in space
without any alteration in form or magnitude (or that ing figures change equally and each
fills
all
the same space
mov when
2
The rigidity postulate in recent a but given geometries, contradictory of it, stated above (the postulate of equal variation) would likewise admit brought back to
its original position).
is
compared by the method of superposition, and would go on quite as well" (Clifford). Of the 'everything two, the former is the simpler postulate and is, moreover, in
figures to be
accordance with what we conceive to be our everyday experi ence. G. B. Halsted contends that Euclid did not miss the rigidity assumption
covers
made In
by
and
is
making it, since he "Magnitudes which can be
justified in not
his assumption 8:
to coincide with one another are equal to one another." the first book, Euclid only once imagines figures to be
moved 1
it
The
so as to
relatively to each other, namely, in proving proposi
" If a straight line meet two straight lines, parallelpostulate is : the two ulterior angles on the same side of it taken to
make
gether less than two right angles, these straight lines, being continually produced, shall at length meet on that side on which are the angles which are less than two right angles." In the various editions of Euclid, different numbers are assigned to the "axioms." Thus, the parallelpostulate is
manuscripts the 5th postulate.
This place
is also assigned to it by compare the various MSS. ) in his edition of Euclid in French and Latin, 1814, and by HEIBEEG and MENGE in their excellent annotated edition of Euclid's works, in Greek and Latin, Leipzig, 1883. Clavius calls it the 13th axiom; Robert Sim
in old
F.
PETBAED (who was
son, the 12th 2
Consult
the
first
to critically
axiom; others (Bolyai
for instance), the llth axiom. Sense of the Exact Sciences,
W. K. CLIFFORD, The Common
" (1) Different things changed equally, and (2) anything which was carried about and brought back to its original position filled tht> same space." 1885, p. 54.
A HISTORY OF MATHEMATICS
72 tion 4
two triangles are equal
:
on this point.
is silent
" Can
sides
and the included
the triangles into coin
to be turned over, but Euclid
may have
cidence, one triangle
two
if
To bring
angle are equal respectively.
it
have escaped his notice that
in plane geometry there is an essential difference between motion of translation and reversion ? " l
Book
on proportions of magnitudes, has been greatly
T.,
admired because of the book
difficult.
its
2 rigour of treatment.
Beginners find
It has been the chief battleground of dis
cussion regarding the fitness of the Elements as a textbook for beginners.
Book X. is
(as also
Books TIL, Till.,
IX.,
omitted from modern school editions.
wonderful book of
all.
XIIL, XIT., XT.) But it is the most
Euclid investigates every possible
variety of lines which can be represented
a and b representing two commensurable species.
lines,
Every individual of every species
V&, by "\ Va and obtains 25 is
incommeiLSu
rable with all the individuals of every other species. enthusiastic in his admiration of this book. a
De
Morgan was 1
2
ENGEL and OTACKEL, p. 8, note. interesting comments on Books V. and
For
VI., see
HANKEL,
pp.
389404. 8
See his articles "Eucleides" in SMITH'S Die. of Greek and Eoman " Irrational 7 ' in the Penny Cyclopaedia Biog. and Myth, and Quantity or in the English Cydop&dia. See also NESSELMANN, pp. 165183. In connection with this subject of irrationals a remark by Dedekind is of See RICHARD DEDEKIND,
Was
und was sollen die Zahlen, points out that all Euclid's constructions of figures could be made, even if the plane were not con tinuous that is, even if certain points in the plane were imagined to be interest.
Braunschweig, 1888, pp.
xii
and
xiii.
sind
He
;
out, so as to give it the appearance of a sieve. in Euclid's constructions would lie between the holes ;
punched
All the points of the
no point
constructions would fall into a hole. The explanation of all this is to be sought in the fact that Euclid deals with certain algebraic irrationals, to the exclusion of the transcendental.
GREECE
73
The main differences in subjectmatter between the Elements and our modern school geometries consist in this the modern works pay less attention to the "Platonic figures," but add theorems on triangles and quadrilaterals inscribed in or cir :
cumscribed about a
circle,
on the centre of gravity of the
tri
angle, on spherical triangles (in general, the geometry of a
spherical surface), and, perhaps, on some of the more recent discoveries pertaining to the geometry of the plane triangle
and
circle.
The main difference as to method, between Euclid and his modern rivals, lies in the treatment of proportion and the development of
sizerelations.
His geometric theory of pro
him to study these relations without reference The Elements and all Greek 'geometry before The theorem that the area Archimedes eschew mensuration. of a triangle equals half the product of its base and its altitude,
portion enables to mensuration.
or that the area of a circle equals TT times the square of the In fact he nowhere finds an radius, is foreign to Euclid.
approximation to the ratio between the circumference and
Another difference
that Euclid, unlike the great majority of modern writers, never draws a line or constructs a figure until he has actually shown the possibility of sucfe con diameter.
is
struction with aid only of the first three of his postulates or of some previous construction. The first three propositions
Book
are not theorems, but problems, (1) to describe an equilateral triangle, (2) from a given point to draw a straight line equal to a given straight line, (3) from the greater of two
of
I.
It is only by straight lines to cut off a part equal to the less. the use of hypothetical figures that modern books can relegate For instance, an all constructions to the end of chapters.
angle
is
imagined to be bisected, before the possibility and One of the most bisecting it has been shown.
method of startling
examples of hypothetical constructions
is
the division
A HISTORY OF MATHEMATICS
74
of a circumference into any desired number of equal parts, given in a modern textbook. This appears all the more star
when we remember that one of the discoveries which name of Gauss immortal is the theorem that, besides
tling,
make
the
n regular polygons of 2 , 3, 5 sides (and combinations therefrom), only polygons whose number of sides is a prime number larger
than
five,
and of the form
p
22
" f
1,
can be inscribed in a
with aid of Euclid's postulates, i.e. with aid of ruler and 1 Is the absence of hypothetical constructions compasses only. circle
commendable
?
If the
aim
aim
is rigour,
we answer
Yes.
If the
ciples,
then we answer, in a general way,
is
emphatically,
adherence to recognized pedagogical prin that, in the transition
from the concrete to the more abstract geometry,
it
often seems
desirable to let facts of observation take the place of abstruse
processes of reasoning. Even Euclid resorts to observation for the fact that his two circles in I., 1 cut each other. 2 Reasoning
More too difficult to be grasped does not develop the mind. 3 over, the beginner, like the ancient Epicureans, takes no interest
him what he has long more apt to interest him
in trains of reasoning which prove to
known; geometrical reasoning
is
when it discloses new facts. Thus, pedagogics may reasona bly demand some concessions from demonstrative rigour. Of the other works of Euclid we mention only the Data, probably intended for students who had completed the Ele ments and wanted drill in solving new problems a lost work ;
on
containing exercises in detecting fallacies; a treatise on Porisms, also lost, but restored by Robert Simson Fallacies,
and Michel Chasles. 1
KLEIN,
2
The Epicureans, says
p. 2.
Proclus, blamed Euclid for proving some things which were evident without proof. Thus, they derided I., 20 (two sides of a triangle are greater than the third) as being manifest even to asses. 8 For discussions of the subject of Hypothetical Constructions, see E. L. RICHARDS in Educat. Review, Yol. in., 1892, p. 34 G. B. HJLLSTED in same journal, Vol. IV., 1893, p. 152. ;
GREECE The
75
period in which Euclid flourished
was the golden era in era This brought forth the two history. most original mathematicians of antiquity, Archimedes and Apollonius of Perga. They rank among the greatest mathe Greek mathematical
maticians of
all
Only a small part of their discoveries
time.
can be described here. Archimedes (287?212
B.C.)
was born
at
He
Cicero tells us he was of low birth.
Syracuse in visited
Sicily.
Egypt and,
perhaps, studied in Alexandria; then returned to his native place, where he made himself useful to his admiring friend and his extraordinary inventive
patron,
King Hieron, by applying
powers
to the construction of warengines,
by which he inflicted on the Romans, who, under Marcellus, were besieging That by the use of mirrors reflecting the sun's rays
gfeat loss
the
he
city.
set
on
fire
the
Eoman
ships,
when they came within bow
shot of the walls, is probably a fiction. Syracuse was taken at the and died in the indiscrim Archimedes length by Eomans, inate slaughter which followed.
The
story goes that, at the
time, he was studying some geometrical diagram drawn in the sand. To an approaching Eoman soldier he called out, " Don't spoil
my
circles,"
The Eoman
but the soldier, feeling insulted, killed him.
general Marcellus,
who admired
his genius, raised
in his honour a tomb bearing the figure of a sphere inscribed in a cylinder. The Sicilians neglected the memory of Archimedes, for
when
Cicero visited Syracuse, he found the
tomb buried
under rubbish.
While admired by ical inventions,
his fellowcitizens
mainly for his mechan
he himself prized more highly his discoveries
in pure science.
Of
special interest to us is his
the Circle. 1
He
proves
first
book on the Measurement of
that the circular area
is
equal to
j
1
A
recent standard edition of his works
188081.
For a
fuller
is
that of HEIBERG, Leipzig,
account of his Measurement of the Circle, see
A HISTOBY OF MATHEMATICS
76
that of a right triangle haying the length of the circumference To find this base for its base and the radius for its altitude. is
He
the next task.
first
finds an upper limit for the ratio of
After constructing an equi vertex in the centre of the circle and
the circumference to the diameter. lateral triangle
with
its
base tangent to the circle, he bisects the angle at the centre and determines the ratio of the base to the altitude of one of its
the resulting right triangles, taking the irrational square root little too small. Xext, the central angle of this right triangle Then the cen is bisected and the ratio of its legs determined.
a
tral angle of this last right triangle is bisected its legs
computed.
and the
This bisecting and computing
is
ratio of
carried on
four times, the irrational square roots being taken every time a little too small. The ratio of the last two legs considered is
> 4673
:
153.
But the shorter
of the legs having this ratio
the side of a regular circumscribed polygon. This leads him to the conclusion that the ratio of the circumference to the is
< 3.
Xext, he finds a lower limit by inscribing regular polygons of 6, 12, 24, 48, 96 sides, finding for each suc In this way he arrives at the cessive polygon its perimeter.
diameter
is
> > 3f,
lower limit 3f Hence the final result, 3 approximation accurate enough for most purposes.
TT
an
Worth noting is the fact that while approximations to TT were made before this time by the Egyptians, no sign of such com putations occur in Euclid and his Greek predecessors.
Why
Perhaps because Greek ideality ex calculation from geometry, lest this noble science
this strange omission?
cluded
all
lose its rigour
veying.
and be degraded to the level of geodesy or sur
Aristotle says that truths pertaining to geometrical
magnitudes cannot be proved by anything so foreign to geomI., pp. 300303, 316, 319 ; LOHUL, IL, pp. 126132 ; Gow, pp. 233H, WEISSEXBORN, Die Berechnung des KreisTJmfanges bei Archi medes und Leonardo Pisano, Berlin, 1894.
CANTOE,
237
;
GBEECE The
etry as arithmetic. in the contention
77
real reason, perhaps,
by some ancient
may
critics that it is
be found
not evident
that a straight line can be equal in length to a curved line in particular that a straight line exists which is equal in length ;
to the circumference.
This involves a real difficulty in geomet
Euclid bases the equality between lines or between areas on congruence. Now, since no curved line, or
rical reasoning.
even a part of a curved line, can be made to exactly coincide with a straight line or even a part of a straight line, no com parisons of length between a curved line and a straight line can be made. So, in Euclid, we nowhere find it given that a
The method employed is equal to a straight line. Greek geometry truly excludes such comparisons according to Duhamel, the more modern idea of a limit is needed to logi curved line in
;
1 cally establish the possibility of such comparisons.
On Euclid
cannot even be proved that the perimeter of a circumscribed (inscribed) polygon is greater (smaller) than
ean assumptions
it
the circumference.
they can see that
Some
writers tacitly resort to observation ;
it is so.
Archimedes went a step further and assumed not only
this,
made the
further assump tion that a straight line exists which equals the circumference in length. On this new basis he made a valued contribution to geometry. No doubt we have here an instance of the usual but, trusting to his ittiuitions, tacitly
course in scientific progress. Epochmaking discoveries, at their birth, are not usually supported on every side by unyield
ing logic ; on the contrary, intuitive insight guides the seeker As further examples of this we instance over difficult places. the discoveries of
and of Maxwell in chain of reasoning by which the truth
Newton
in mathematics
The complete physics. of a discovery is established is usually put together at a later period. i
G. B. HALSTED in Tr. Texas
Academy of
Science, L, p. 96.
A HISTORY OF MATHEMATICS
T8 Is not the
individual
same course
mind
?
We
of
advancement observable
grasping the reasons for them. Xor the young mind should, from the start, grasp them
all.
in the
truths without fully is it always best that
first arrive at
make the
effort to
In geometrical teaching, where the reasoning
too hard to be mastered, if observation can conveniently student cannot wait until he has assist, accept its results.
is
A
mastered limits and the calculus, before accepting the truth that the circumference is greater than the perimeter of an inscribed polygon.
Of
Archimedes prized most highly those in his book on the Sphere and Cylinder. In this he uses the all his discoveries
celebrated statement, "the straight line is the shortest path between two points," but he does not offer this as a formal definition of a straight line. 1 Archimedes proves the new
theorems that the surface of a sphere is equal to four times a great circle; that the surface of a segment of a sphere is equal to a circle whose radius is the straight line drawn from the vertex of the segment to the circumference of its basal circle ; that the volume and the surface of a sphere are  of the volume and surface, respectively, of the cylinder circum scribed about the sphere. The wish of Archimedes, that the
upon his tomb, was the Roman Marcelius. general by Archimedes further advanced solid geometry by adding to the five "Platonic figures," thirteen semiregular solids, each
figure for the last proposition be inscribed
carried out
bounded by regular polygons, but not
same kind. To elementary geometry belong also his fifteen Lemmas* Of the " Great Geometer," Apollom'us of Perga, who flour all
of the
ished about forty years after Archimedes, and investigated the properties of the conic sections, we mention, besides his 1
CAJTTOE,
2
Consult Grow,
I.,
298. p.
232
;
CANTOE,
L
s
p. 298.
GBEECB celebrated
work on
Conies, only
79 a lost work on Contacts,
which Yieta and others attempted to restore from certain lemmas given by Pappus. It contained the solution of the celebrated "Apollonian Problem": Given three circles, to find a fourth, which shall touch the three. Even in modern times this problem has given stimulus toward perfecting geo metric methods. 1
Archimedes, and Apollonius, the eras of Greek But little is geometric discovery reach their culmination. known of the history of geometry from the time of Apollonius "With. Euclid,
to the beginning of the Christian era.
In this interval
falls
Zenodorm, who wrote on Figures of Equal Periphery. This book is lost, but fourteen propositions of it are preserved by
Pappus and circle
also
by Theon.
Here are three of them:
a
The
has a greater area than any polygon of equal periphery,"
"Of
polygons of the same number of sides and of equal " Of all solids periphery the regular is the greatest," having surfaces equal in area, the sphere has the greatest volume."
Between 200 and 100 B.C. lived Hypsides, the supposed author of the fourteenth book in Euclid's Elements. His treatise on Risings is the earliest Greek work giving the division of the 360 degrees after the manner of the Babylonians. HipparcJius of Niceea in Bithynia, the author of the famous
circle into
theory of epicycles and eccentrics,
is
the greatest astronomer
Theon
of Alexandria tells us that he originated the science of trigonometry and calculated a " table of chords " of antiquity.
Hipparchus took astronomical observations between 161 and 126 B.C. A writer whose tone is very different from that of the great
in twelve books (not extant).
writers of the First Alexandrian School
was Heron of Alexan
1 Consult E. SCHILKE, Dte Losungen und Erweiterungen des Apollonischen Beruhrungsproblems, Berlin, 1880.
A HISTORY OF MATHEMATICS
80
dria, also called
Heron the Elder. 1
As he was a
practical
not surprising to find little resemblance between surveyor, his writings and those of Euclid or Apollonius. it is
Heron was a pupil
of Ctesibius,
who was
celebrated for his
mechanical inventions, such as the hydraulic organ, waterclock, and catapult. It is believed by some that Heron was a son of Ctesibius.
Heron's invention of the
eolipile
and a curious
mechanism, known as "Heron's Fountain," display talent of the same order as that of his master.
Most
regarding his writings.
Great uncertainty exists him to be
authorities believe
the author of a work, entitled Dioptra, of which three quite Marie 2 thinks that the dissimilar manuscripts are extant. is the work of a writer of the seventh or eighth cen tury A.D., called Heron the Younger. But we have no reliable evidence that a second mathematician by the name of Heron
Dioptra
3 really existed.
A
reason adduced by Marie for the later it is the first work which
origin of the Dioptra is the fact that
contains the important formula for the area of a triangle, ex pressed in terms of the three sides. Now, not a single Greek
writer cites this formula
;
hence he thinks
it
improbable that
the Dioptra was written as early as the time of Heron the Elder. This argument is not convincing, for the reason that
only a small part of Greek mathematical literature of this period has been preserved. The formula, sometimes called"Heronic formula," expresses the triangular area as follows: f
q
V where
to
a,
2222
+b+c
a+
b} c are the sides.
b
c
a
+c
The proof
b
given,
though laborious,
1 According to CANTOR, I, 363, he flourished about 100 MARIE, I., 177, about 155 B.C. 2 MARIE, L, 180. 8 CANTOR, I,, 367.
B.C.
;
according
GREECE is
81
exceedingly ingenious and discloses no
little
mathematical
1
ability.
"
Dioptra," says Venturi,
modern
theodolites.
yards long with
" were instruments resembling our
The instrument consisted
little plates
rested upon a circular disc.
of a rod four
This
at the ends for aiming.
The rod could be moved horizon
and also vertically. By turning the rod around until stopped by two suitably located pins on the circular disc, the surveyor could work off a line perpendicular to a given direc tion. The level and plumbline were used." 2 Heron explains with aid of these instruments and of geometry a large num tally
ber of problems, such as to find the distance between two points of which one only is accessible, or between two points
which are visible, but both inaccessible from a given point to run a perpendicular to a line which cannot be approached; to find the difference of level between two points to measure ;
;
the area of a field without entering it. The Dioptra discloses considerable mathematical ability and familiarity with the writings of the authors of the classical period.
Heron had read Hipparchus and he wrote a commen
3 tary on Euclid.
Nevertheless the character of his geometry not Grecian, but Egyptian. Usually he gives directions and He gives formulae for computing the rules, without proofs. is
area of a regular polygon from the square of one of its sides j this implies a knowledge of trigonometry. Some of Heron's formulae point to an old Egyptian origin.
Thus, besides the
formula for a triangular area, given above, he gives ^"jf 03 x , & A
which bears a striking likeness 1
375 2
For the ;
Gow,
proof, see Dioptra (Ed.
to the formula
ag
ai ;" %
HULTSCH), pp. 235237
;
x
G
~T
2
CAHTOB,
p. 281.
CANTOR, L, 382.
!
See TANNEEY, pp. 166181.
*,
L,
A HISTOKY OF MATHEMATICS
82
for finding the area of a quadrangle, found in the
Edfu inscrip
There are points of resemblance between Heron and Ahmes. Thus, Ahmes used unitfractions exclusively Heron tions.
;
used them oftener than other fractions. theories of
Ahmes were
demonstrated by the oldest
That the arithmetical
not forgotten at this time
Akhmim
is
also
papyrus, which, though the
extant textbook on practical Greek arithmetic, was
probably written after Heron's time.
Like
Ahmes and
the
Edfu, Heron divides complicated figures into simpler ones by drawing auxiliary lines; like them he displays par ticular fondness for the isosceles trapezoid. priests at
The
Heron
satisfied a practical
want, and for that reason were widely read. find traces of them in Borne, in the Occident during the Middle Ages, and even in India. writings of
We
VI.
The Second Alexandrian School
of
With
the absorption
Boman Empire and
with the spread of Egypt Alexandria became a commercial as well as great Christianity, intellectual centre. Traders of all nations met in her crowded into the
while in her magnificent library, museums, and lecturerooms, scholars from the East mingled with those of the West. streets,
Greek thinkers began to study the Oriental literature and The resulting fusion of Greek and Oriental
philosophy.
philosophy
The study
led to XeoPythagoreanism and ISTeoPlatonism. of Platonism
and Pythagorean mysticism led to a
revival of the theory of numbers. This subject became again a favourite study, though geometry still held an important
This second Alexandrian school, beginning with the made famous by the names of Diophantus, Claudius Ptolemseus, Pappus, Theon of Smyrna, Theon place.
Christian era, was
of Alexandria, lamblichus, Porphyrius, Serenus of Antinoeia,
Menelaus, and others. Menelaus of Alexandria lived about 98
A.D., as
appears from
GBEECE
83
two astronomical observations taken by him and recorded in Valuable contributions were made by him to spherical geometry, in his work, Sphverica, which is extant in Hebrew and Arabic, but lost in the original Greek. He gives the theorems on the congruence of spherical triangles and de
the Almagest. 1
scribes their properties in
much
the same
way
as Euclid treats
He gives the theorems that the sum of the plane triangles. three sides of a spherical triangle is less than a great circle, and that the sum. of three angles exceeds two right angles. Celebrated are two theorems of his on plane and spherical tri The one on plane triangles is that " if the three sides angles. be cut by a straight
line,
segments which have no uct of the other three/ this proposition,
7
known
the product of the one set of three extremity is equal to the prod
common
The illustrious Lazare Carnot makes as the " lemma of Menelaus," the base
of his theory of transversals. 3
The corresponding theorem
for
spherical triangles, the socalled "'rule of six quantities," is obtained from the above by reading " chords of three segments " doubled," in place of three segments." Another fundamental theorem in modern geometry (in the
theory of harmonics) Antinoeia If from
is
D we
:
H
the following ascribed to Serenus of draw DF, cutting the triangle ABG,
on it, so that DE DF= HE HF, and if we and choose draw the line AH. then every transversal through D, such as :
:
1
CANTOR, L, 412. For the history of the theorem, see M. CHASLES, Geschichte der GeomAus dem Franzosischen ubertragen durch DR. L. A. SOHNCKE. etrie. Hereafter we quote this work as Halle, 1839, Xote VI., pp. 295299. CHASLES. A recent French edition of this important work is now easily " was well Chasles points out that the " lemma of Menelaus obtainable. known during the sixteenth and seventeenth centuries, but from that time, for over a century, it was fruitless and hardly known, until finally 2
Carnot began his researches. theorem.
Carnot, as well as Ceva, rediscovered the
A HISTORY OF MATHEMATICS
84
DG,
will be divided
by
AH so
AntincBia (or Antinoupolis) in
that
DK\ DG = JK: JG. As
Egypt was founded by Emperor Hadrian in 122 A.D., we have an upper limit of Serenus.
he
is
1
for the date
The
fact that
quoted by a writer of
tne fiftk or sixth century sup plies us with a lower limit.
A
central position in the history of ancient astronomy
is
occupied by Claudius Ptolemceus. Nothing is known of his personal history except that he was a native of Egypt and flourished in Alexandria in 139 A.D.
The
chief of his works
are the Syntaxis Mathematica (or the Almagest} as the Arabs called it) and the Geographica. This is not the place to de scribe the "Ptolemaic
System"; we mention Ptolemseus because
of the geometry and especially the trigonometry contained in the Almagest. He divides the circle into 360 degrees; the
diameter into 120 divisions, each of these into 60 parts, which are again divided into 60 smaller parts. In Latin, these parts
were called partes minutce primce and partes mmutce secundce. 2 Hence our names "minutes" and "seconds." Thus the Baby lonian sexagesimal system, known to Geminus and Hipparchus, had by this time taken firm root among the Greeks in Egypt.
The foundation of trigonometry had been laid by the Hipparchus.
He
form.
illustrious
Ptolemseus imparted to it a remarkably perfect calculated a table of chords by a method which
seems original with him. After proving the proposition, now appended to Euclid, VI. (D), that " the rectangle contained by the diagonals of a quadrilateral figure inscribed in a circle is equal to both the rectangles contained by its opposite sides," he shows how to find from the chords of two arcs the chords of 1
3
J. L.
HEIBERG
CANTOR,
I., p,
in jBibliotheca
416.
Mathematica, 1894,
p. 97.
GREECE
85
sum and
difference, and from the chord of any arc that These theorems, to which he gives pretty proofs, 1 It goes without say are applied to the calculation of chords. ing that the nomenclature and notation (so far as he had any)
their
of its half.
were entirely different from those of modern trigonometry. In " chord of double the arc." place of our sine," he considers the *
Thus, in his table, the chord 21 21 12 is given for the arc 20 30'. Eeducing the sexagesimals to decimals, we get for the
chord .35588.
Its half, .17794,
or the half of 20
is
seen to be the sine of 10
15',
30'.
More complete than in plane trigonometry is the theoretical 2 Ptolemy exposition of propositions in spherical trigonometry. The fact that starts out with the theorems of ilenelaus. trigonometry was cultivated not for its own sake, but to aid astronomical inquiry, explains the rather startling fact that spherical earlier
trigonometry came to exist in a developed state
than plane trigonometry.
Of particular interest to us is a proof which, according to Proclus, was given by Ptolemy to Euclid's parallelpostulate.
The
critical part of
the proof
is
as follows: If the straight
lines are parallel, the interior angles [on the
same
transversal] are necessarily equal to are not two right angles. For 717 ,
less parallel fore, /???,
than
0,
77$
and, there
whatever the sum of the angles whether greater or less than 778,
side of the
j

^
5
1
^
must be But the sum of the four 777. cannot be more than four right angles, because they are two
two right the
sum
angles, such also
of the angles #77,
pairs of adjacent angles. 1
2
8
3
The untenable point
Consult CANTOB, Consult CANTOE,
Gow,
p. 301.
I., I.,
in this proof
417, for the proofs. Gow, p. 297. p. 420 ;
A HISTORY OF MATHEMATICS
86 is
the assertion that, in case of parallelism, the sum of the on one side of the transversal must be the same
interior angles
as their
sum on
the other side of the transversal.
Ptolemy
a long line of geometers who dur ing eighteen centuries vainly attempted to prove the parallelpostulate, until finally the genius of Lobatchewsky and Bolyai
appears to be the
first of
dispelled the haze
cians
which obstructed the vision of mathemati
and led them
reason
to see with unmistakable clearness the
such proofs have been and always will be
why
Por 150 years
An
after Ptolemy, there appeared
futile.
no geometer
work, entitled Cestes, by Sextus Julius African as, applies geometry to the art of war. Pappus (about 300 or 370, A.D.) was the last great mathematician of the Alexandrian school. Though inferior in genius to Archi of note.
unimportant
medes, Apollonius, and Euclid, who wrote five centuries earlier, Pappus towers above his contemporaries as does a lofty peak
above the surrounding plains. books
;
the
first
Of
his several
the only one extant. and parts of the second are
matical Collections
works the Mathe It
is
was in
eight
now missing. The
object of this treatise appears to have been to supply geometers with a succinct analysis of the most difficult mathematical
works and lemmas.
to
It
information
facilitate the is
it
study of these by explanatory invaluable to us on account of the rich
gives
on various treatises by the foremost
Greek mathematicians, which are now
lost.
Scholars of the
century considered it possible to restore lost works from the resume given by Pappus alone. Some of the theorems last
are doubtless original with Pappus, but here it is difficult to speak with certainty, for in three instances Pappus copied
theorems without crediting them to the authors, and he may have done the same in other cases where we have no means of ascertaining the real discoverer.
Of elementary propositions
of special interest
and probably
GREECE own, we^mention the following
ins
87
:
(1)
The
centre of inertia
that of another triangle whose vertices (gravity) of a triangle the first and divide its three sides in lie upon the sides of is
the same ratio
;
(2)
solution of the problem to
draw through
three points lying in the same straight line, three straight l lines which shall form a triangle inscribed in a given circle ;
he propounded the theory of involution of points
(4)
the
(3) line connecting the opposite extremities of parallel diameters ;
circles passes through the point of theorem contact <'a suggesting the consideration of centres of the problem, to of similitude of two circles) (5) solution
of
two externally tangent
;
find a parallelogram
whose
sides are in a fixed ratio to those of
a given parallelogram, while the areas of the two are in another This resembles somewhat an indeterminate prob
fixed ratio.
lem given by Heron, to construct two rectangles in which the 2 sums of their sides, as well as their areas, are in given ratios. It remains for us to name a few more mathematicians. Theon of Alexandria brought out an edition of Euclid's Ele ments, with notes his commentary on the Almagest is valuable on account of the historical notes and the specimens of Greek arithmetic found therein. Theon's daughter Hypatia^ a woman;
renowned for her beauty and modesty, was the last Alexandrian teacher of reputation. Her notes on Diophantus and Apollonius have been lost. Her tragic death in 415 A.D. is vividly described in Kingsley's Hypatia* From now on Christian theology absorbed men's thoughts.
,
1
"The
become
problem, generalized by placing the points anywhere, has by its difficulty, partly by the names of geom
celebrated, partly
especially by the solution, as general as it was of 16, Ottaiano of Naples." CHASLES, p. 41; 2 CANTOS, I., 464. Chasles gives a history of the problem in note 8 The reader may be interested in an article by G. VALENTIN, "Die
eters
who
solved
simple, given
it,
and
by a boy
XL
Frauen in den exakten Wissenschaften," Bibliotheca Mathematica, 1895, pp. 6576.
A HISTORY OP MATHEMATICS
88
In Alexandria Paganism disappeared and with it Pagan learn The XeoPl atonic school at Athens struggled a century ing. Proclus, Isidonis, and others endeavoured to keep up "the golden chain of Platonic succession. 77 Proclus wrote a commentary on Euclid that on the first book is extant and is longer.
;
of great historical value.
A
pupil of Isidorus, Damascius of
Damascus (about 510 A.D.) is believed by some to be the author of Book XV. of Euclid's Elements. The geometers of the last 500 years, with the possible exception of Pappus, lack creative power; they are commenta tors rather than originators.
The
Greek geometry are wonderful clearness and definiteness of concepts, and an exceptional logical rigour of conclusions. We have salient features of
:
A
(1)
encountered occasional flaws in reasoning; but when we com pare Greek geometry in its most complete form with the best that the Babylonians, Egyptians, Eomans, Hindus, or the geometers of the Middle Ages have brought forth, then it
must be admitted that not only
in rigour of presentation, but
also in fertility of invention, the geometric
towers above
all
mind
of the Greek
others in solitary grandeur.
(2) A complete absence of general principles and methods. For example, the Greeks possessed no general method of drawing tangents. In the demonstration of a theorem, there
were, for the ancient geometers, as many different cases re quiring separate proof as there were different positions for the 1
"It is one of the greatest advantages of the more modern geometry over the ancient that through the considera tion of positive and negative qiiantities it embraces in one expression the several cases which a theorem can present in lines.
respect to the various relative positions of the separate parts 1
Take, for example, EUCLID,
UL,
35.
89
main problems and the which are the subjectmatter of 83 special cases, two de books sectione determinates the theorems in (of Pappus), of the figure.
Titus today the nine
numerous
which can be solved by one single "If we compare a mathematical problem with a huge rock, into the ulterior of which we desire to penetrate, then the work of the Greek mathematicians appears to us constitute only one problem 1
equation."
who, with chisel and ham with indefatigable perseverance, from without, to mer, begins crumble the rock slowly into fragments the modern mathe like that of a vigorous stonecutter
;
matician appears like an excellent miner, who first bores through the rock some few passages, from which he then bursts
it
into pieces with one powerful blast,
light the treasures within."
and brings to
2
EOME Although the Eomans excelled in the science of government and war, in philosophy, poetry, and art they were mere imita In mathematics they did not even rise to the desire for tors. imitation.
If
we except the
period of decadence, during
which the reading of Euclid began, we can say that the classical Greek writers on geometry were wholly unknown in Borne.
A
science of geometry with definitions, postulates, axioms, practical geometry, rigorous proofs, did not exist there. like the old Egyptian, with empirical rules applicable in sur
A
veying, stood in place of the Greek science.
prepared by Roman surveyors,
have come down to
us.
"
As
Practical treatises
called agrimensores or gromatid,
regards the geometrical part of
1
CHASLES,
2
HEEMANN HANKEL, Die Entwickelung
p. 39.
Jahrhunderten, Tubingen, 1884,
p. 9.
der Maihematik in den letzten
A HISTORY OF MATHEMATICS
90
these pandects, which treat exhaustively also of the juristic and purely technical side of the art, it is difficult to say whether the crudeness of presentation, or the paucity and faultiness
of the contents
more strongly repels the
reader.
The presentation beneath the notice of criticism, the termi nology vacillating; of definitions and axioms or proofs of the are not formu prescribed rules there is no mention. The rules is
lated;
the reader
is
left
to
abstract
them from numerical
The total examples obscurely and inaccurately described. were Roman thousands the as is gromatici though impression of years older than Greek geometry, and as though the deluge 1 Some of their rules were were lying between the two." the Etruscans, but others are identi probably inherited from the latter is that for finding ?? the area of a triangle from its sides [the " Heronic formula ] 2 and the approximate formula, ^f a for the area of equilateral cal
with those of Heron.
Among
,
But the latter area was triangles (a being one of the sides). 2 a2 the first of also calculated by the formulas 4 (a 4 a) and J
which was unknown to Heron.
Probably
4
,
a 2 ^as derived
from an Egyptian formula. The more elegant and refined methods of Heron were unknown to the Romans. The grom atici
considered
it
sometimes
sufficiently accurate to
determine
the areas of cities irregularly laid out, simply by measuring their circumferences. 2 Egyptian geometry, or as much of it
Romans thought they could use, was imported at the time of Julius Caesar, who ordered a survey of the whole as the
empire to secure an equitable mode of taxation. From early times it was the Roman practice to divide land into rectan
Walls and streets were parallel, gular and rectilinear parts. enclosing squares of prescribed dimensions. This practice 1 HANSEL, pp. 295, 296. For a detailed account of the agrimensores, consult CANTOS, I., pp. 521538. 2 HANSEL, p. 297.
KOHE simplified matters
sary amount
91
immensely and greatly reduced the neces
Approximate formulae demands of answered all ordinary precision. Caesar reformed the calendar, and for this undertaking drew from Egyptian learning. The Alexandrian astronomer 8osigenes
was
of geometrical knowledge.
enlisted for this task.
Among Roman names
identi
with geometry or surveying, are the following: Marcus Terentius Varro (about 11627 B.C.), Sextus Julius Frontinus (in
fied
70 A.B. praetor in Rome), Ifartiamts Mineus Felix Capella rn at Carthage in the early part of the fifth century), Magnus Aurelius Cassiodorius (born about 475 A.D.). Vastly superior to any of these were the Greek geometers belonging to the period of decadence of Greek learning. It is a remarkable fact that the period of political humilia tion,
marked by the
fall of the
Western Roman Empire and is the period during which
the ascendancy of the Ostrogoths,
the study of Greek science began in Italy. The compila tions made at this time are deficient, yet interesting from the fact that, down to the twelfth century, they were the only Toresources of mathematical knowledge in the Occident.
most among these writers
At
is
Anieius
Manlius
Severinus
a favourite of King Theodorie 5 he was later charged with treason, imprisoned, and While in prison he wrote On the Con finally decapitated. Boetkius (480?524),
first
of Philosophy. Boethius wrote an Institutio Arithmetica (essentially a translation of the arithmetic of Kico
solations
machus) and a Geometry. The first book of his Geometry is an extract from the first three books of Euclid's Elements, with the proofs omitted. It appears that Boethius and a num ber of other writers after
him were somehow
led to the belief
that the theorems alone belonged to Euclid, while the proofs were interpolated by Theon hence the strange omission of all ;
demonstration.
The second book
in the Geometry of Boethius
92
A HISTORY OF MATHEMATICS
consists of an abstract of the practical geometry of Frontinus,
the most accomplished of the gromatici.
Notice that, imitating Mcomachus, Boethius divides the
mathematical sciences into four sections, Arithmetic, Music,
Geometry, Astronomy.
He
first
designated them by the word
quadruvium (four pathways). This term was used extensively during the Middle Ages. Gassiodorius used a similar figure, the four gates of science. his Origines,
groups
Isidorus of Carthage (born 570), in the four embraced
all sciences as seven,
by the quadruvium and three (Grammar, Ehetoric, Logic) which constitute the trivium (three pathways).
MIDDLE AGES
ARITHMETIC AND ALGEBKA Hindus
Soox
after the decadence of
Greek mathematical research,
another Aryan race, the Hindus, began to display brilliant mathematical power. ]>Tot in the field of geometry, but of arithmetic and algebra, they achieved glory. In geometry
they were even weaker than were the Greeks in algebra. The subject of indeterminate analysis (not within the scope of this
was conspicuously advanced by them, but on this point they exerted no influence on European investigators, for the reason that their researches did not become known history)
in the Occident until the nineteenth century.
India had no professed mathematicians
;
we are To them,
the writers
about to discuss considered themselves astronomers.
mathematics was merely a handmaiden to astronomy. In view of this it is curious to observe that the auxiliary science is after all the only one in which they won real dis tinction, while in their pet pursuit of astronomy they displayed an inaptitude to observe, to collect facts, and to make inductive investigations.
It is an unpleasant feature about the
Hindu mathematical
handed down to us that rules and results are expressed ia verse and clothed in obscure mystic language. To him who treatises
A HISTORY OF MATHEMATICS
94
already understands the subject such verses may aid the memory, but to the uninitiated they are often unintelligible.
Usually proofs are not preserved, though Hindu mathemati cians doubtless reasoned out all or most of their discoveries. It is certain that portions
Greek
An
origin.
Hindu mathematics
of
of the relation between
Hindu
are of
the tracing After and Greek thought.
interesting but difficult task
is
Egypt had become a Roman province, extensive commercial relations sprang up "with Alexandria. Doubtless there was considerable interchange of philosophic and scientific know The Hindus were indebted to Heron, Diophantus and ledge. Ptolemy. They were borrowers also from the Chinese.
At present mathematics. science in its
we know very little of the growth of Hindu The few works handed down to us exhibit the complete state only. The dates of all important
works but the
first are
at Bakhshali, in
anonymous
well fixed.
arithmetic, supposed,
verses, to date
The document
In 1881 there was found
northwest India, buried in the earth, an
from the
peculiarities of its
from the third or fourth century after Christ. is of birch bark and is an in
that was found
complete copy, prepared probably about the eighth century, of an older manuscript. 1 The earliest Hindu astronomer known to us is Aryabhatta,
He A,D., at Pataliputra, on the upper Ganges. the author of the celebrated work, entitled Aryalhattiyam, the third chapter of which is given to mathematics. 2 About born in 476 is
one hundred years later flourished Brahmagvpta, born 598, who in 628 wrote his BrahmasphutOrsiddhanta (" The revised system of Brahma"), of which the twelfth and eighteenth chapters 1
CANTOR, I., pp. 598, 613615. See also The Bakhshali Manuscript, by RUDOLF HOERITLE in the Indian Antiquary, XVII. 3348 and
edited
275279, Bombay, 1888. 2 Translated by L. EODET in Journal Asiatigue, 1879,
,
se"rie 7,
T.
95
Bxsx>rs
1 belong to mathematics. Important is an elementary work by ^fabavira (9th century ?) ; then come, Cridkara, who wrote a
GanitOrsara (" Quintessence of Calculation"), and Padmanabka,
the author of an algebra.
but
little
The
science seems to have for a
progress since
work
made
entitled
Brahmagupta, Siddhantadromani ("Diadem of an Astronomical System"), written by Bhaskam Acarya in 1150, stands little higher than Brahmagupta's Siddhanta, written over five hundred years
The two most important mathematical chapters in Bhaskara's work are the Lilavati ("the beautiful," i.e. the noble science) and Vigaganita (" rootextraction "), devoted to earlier,
j
arithmetic and algebra.
From
this time the spirit of research
was extinguished and no great names appear. Elsewhere we have spoken of the Hindu's clever use of the principle of local value tion.
tion,
of
and of the zero in arithmetical nota
We now give
an account of Hindu methods of computa which were elaborated in India to a perfection undreamed
by
earlier nations.
The information handed down
to our
derived partly from the Hindu works, but mainly from an arithmetic written by a Greek monk, Sfaximus Planudes,
time
is
who
lived in the first half of the fourteenth century, and
who
avowedly used Hindu sources. To understand the reason why certain modes of computation were adopted^ we must bear in mind the instruments at the disposal of the
Hindus in
their written calculations.
They
1 Translated into English by H. T. COLEBROOKE, London, 1817. This celebrated Sanskrit scholar translated also the mathematical chapters of the Siddhantadromani of Bhaskara. Colebrooke held a judicial office in
and about the year 1794 entered upon the study of Sanskrit, that he might read Hindu lawbooks. From youth fond of mathematics, he also began to study Hindu astronomy and mathematics ; and, finally, by his translations, demonstrated to Europe the fact that the Hindus in had previous centuries had made remarkable discoveries, some of which India,
been wrongly ascribed to the Arabs.
A HISTORY OF MATHEMATICS
96
wrote " with a cane pen upon a small blackboard with a white^ thinlyliquid paint which made marks that could be easily erased, or
with red
upon a white tablet, less than a foot square, strewn flour, on which they wrote the figures with a small
the figures appeared white on a red ground." 1 To be legible the figures had to be quite large, hence it became necessary to devise schemes for the saving of space. This stick, so that
was accomplished by erasing a digit as soon as it had done The Hindus were generally inclined to follow the motion from left to right, as in writing. Thus in adding 254 and 663, they would say 2 + 6 = 8, 5 } 6 = 11, which changes 8 to 9, 4 + 3 = 7. Hence the sum 917. In subtraction, they had two methods when "borrowing" became necessary. Thus, in 51 28, they would say 8 from 11 = 3, 2 from 4 = 2; or they would say 8 from 11 = 3, 3 from 5 = 2. In multiplication several methods were in vogue. Sometimes they resolved the multiplier into its factors, and then multi plied in succession by each factor. At other times they would resolve the multiplier into the sum or the difference of two numbers which were easier multipliers. In written work, a
its service.
multiplication, such as 5x57893411, was done thus: 5x5=25, which was written down above the multiplicand 5 x 7 = 35 ;
;
adding 3 to 25 gives 28 erase the 5 and in its place write 8. We thus have 285. Then, 5x8 = 40 4 5 = 9; replace 5 by 9, and we have 2890, etc. At the close of the operation the ;
;
work on the
tablet appeared
somewhat
+
as follows
:
289467055 57893411
When Hindu
5
the multiplier consisted of several digits, then the by Hankel (p. 188), in case of
operation, as described 1
HANKEL,
p. 186.
HINDUS 324 x 753, was as follows 2259
^324 753
:
97
Place the lefthand digit of the
multiplier, 324, over units' place in the multiplicand;
3x7 = 21, write by 22; 3x3 = 9.
this
At
down
3 x 5
=
15; replace 21 the stage appearance of the IsText, multiplicand is ;
this
the work is as here indicated. moved one place to the right. 2 x 7 = 14 in the place where the 14 belongs we already have 25, the two together give 39, ;
24096 ^324 <53
written in place of 25 2 x 5 = 10 add 10 and write 409 in place of 399; 2x3 = 6. The adjoining figure shows the work at this stage.
which
is
;
;
to 399,
We
begin the third step by again moving the multiplicand to the right one place 4 x 7 28 add this to 09 and write
=
;
;
down 37
in its place, etc.
243972 324
This method, employed by Hindus even at the pres
^
ent time, economizes space remarkably well, since only a few of all the digits used appear on the tablet at any
one moment. tablets
It was, therefore, well
and coarse
adapted to their small
The method
is a poor one, if the pencils. calculation is to be done on paper, (1) because we cannot readily and neatly erase the digits, and (2) because, having plenty of paper, it is folly to save space and thereby un
necessarily complicate the process by performing the addition of each partial product, as scon as formed. Nevertheless, we find that the early Arabic writers, unable to
Hindu
process, adopted it
and showed how
improve on the can be carried
it
out on paper, viz., by crossing out the digits (instead of eras 1 ing them) and placing the new digits above the old ones. Besides these, the Hindus had other methods, more closely
resembling the processes in vogue at the present time. Thus a tablet was divided into squares like a chessboard. Diagonals
were drawn.
The
multiplication of 12 1
HLNKEL,
p. 188.
x 735
= 8820
is ex
A HISTORY OF MATHEMATICS
98
Mbited in
tlie
The manuscripts extant adjoining diagram. give no detailed information regarding the 1
Hindu
It seerns that process of dmsion. the partial products were deducted by erasing digits in the dividend and replac
ing them by the the subtraction.
new ones resulting from The above scheme of
multiplication was known to the Arabs and may not be of Hindu origin. The Hindus used an ingenious process (not of Indian It origin) for testing the correctness of their computations. rests on the theorem that the sum of the digits of a number,
divided by 9, gives the same remainder as does the number J? divided by 9. The process of " casting out the 9"s was
itself,
more
serviceable to the
Hindus than
it is
to us.
Their custom
of erasing digits and writing others in their places made it much more difficult for them to verify their results by review
ing the operations performed. At the close of a multiplication a large number of the digits arising during the process were Hence a test which did not call for the examination erased. of the intermediate processes was of service to them. In the extant fragments of the Bakhshdll arithmetic, a
knowledge of the processes of computation is presupposed. In fractions, the numerator is written above the denominator without a dividing line. with the denominator 1. part
is
of our
Integers are written as fractions In mixed expressions the integral 1 written above the fraction. Thus, 1 1. In place 3 they used the word phalam, abbreviated into pha.
=
=
Addition was indicated by yu, abbreviated from yuta. Xumbers to be combined were often enclosed in a rectangle.
means f
Thus, pha 12 1
CANTOB, L,

p.
f
=
till.
12.
An unknown
HINDUS sunya, and
99
by a heavy word for the and is empty," here likewise is which a dot. This double zero, represented by use of the word and dot rested upon the idea that a position is
quantity
is
"
is
designated
"empty" empty
"
not
if
out
filled
so long as the
thus
u
The word sunya means
dot.
It is also to
number
be considered
to be placed there has not
been ascertained. 1
The Bakhshali
arithmetic contains problems of which some by a sort of false position.
are solved by reduction to unity or
EXAMPLE B
gives twice as much as A, C three times as much four times as much as C ; together they give 132 ; :
as B,
B
did A give? Take 1 for the unknown A = 1, B = 2, C = 6, D = 24, their sum = 33. 132 by 33, and the quotient 4 is what A gave.
how much
(swnt/a),
then
The method
Divide
we have encountered before With them it was an instinctive
of false position
the early Egyptians. procedure; with the Hindus
among
it
had risen to a conscious method.
Bhaskara uses
it, but while the Bakhshali document preferably 1 the assumes as unknown, Bhaskara is partial to 3. Thus,
if
a certain number
is
taken
fivefold,
subtracted, the remainder divided original
A
is
3,
10,
\ of the product be and \ of the ^, %,
and
obtained.
then you get 15, 10, 2 3 if) = 48, the answer.
Choose
ber ?
Then
number added, then 68
by
1,
and 1
What
is
the
num
+ f f f + f = J.
(68 *favourite method of solution is that of inversion.
laconic brevity, Aryabhatta describes it thus:
With
"Multiplica
tion becomes division, division becomes multiplication; what was gain becomes loss, what loss, gain ; inversion." Quite
from this in style is the following problem of " Beautiful maiden Aryabhatta, which illustrates the method with beaming eyes, tell me, as thou understandst the right different
:
1
CAHTOR, L, pp. 613615.
*
CANTOR,
I,, p.
618.
A HISTORY OF MATHEMATICS
100
of inversion, which is the number which multiplied by then increased 3, by f of the product, divided by 7, diminished of the quotient, multiplied by itself, diminished by 52, by by ^
method
extraction of the square root, addition of 8, and division by 10, " The gives the number 2 ? process consists in beginning with
2 2 and working backwards. Thus, (2 10  8) + 52 = 196, V196 14, and 14 f T $ ~ 3 = 28, the answer. 1 Here is *
=

another example taken from the Lilamti: "The square root number of bees in a swarm has flown out upon a
of half the
jessaminebush, f of the whole swarm has remained behind; one female bee Hies about a male that is buzzing within a lotusflower into which he was allured in the night by its sweet odour, but bees." 2
is
now imprisoned
Answer
72.
The
in
it.
Tell
me
the
pleasing poetic garb in
number
of
which arith
metical problems are clothed is due to the Hindu practice of writing all schoolbooks in verse, and especially to the fact that these problems, propounded as puzzles, were a favourite " These social amusement. problems are Says Brahmagupta :
proposed simply for pleasure; the wise man can invent a thousand others, or he can solve the problems of others by the rules given here. As the sun eclipses the stars by his brilliancy, so the
man
of knowledge will eclipse the fame of others in
assemblies of the people still
more
if
if
he proposes algebraic problems, and
he solves them."
The Hindus were
familiar with the Eule of Three, with the
computation of interest (simple and compound), with alliga tion, with the fountain or pipe problems, and with the sum mation of arithmetic and geometric series. Aryabhatta a 16 year old applies the Rule of Three to the problem one 20 and 32 what costs costs slave nishkas, girl years old? says that
it is
treated by inverse proportion, since "the value
,
p. 618.
101
HIHDUS
ml
creatures
of
is 1
regulated bj their of square
The age/* the older being the cheaper. to the Hindus. roots was f
This was dene
aid of the formulas
contributions to Algebra.
The Hindus Addition was
as ia
indicated simply by
M^
by placing a dot OTer the
algebra;
the factors, subtrahend; mtdtiplioation, by putting after "the product"; division the abbreviation of the word
by placing the diTisor beneath the dividend square root, by before the (irrational), writing tax, from the word was called by quantity. The unknown quantity In ease of several unknown quantities* he vdvatt&yat. ;
rfsHke Diephantas, a distinct name and symbol to eack kno^^Mraantities after the first were assigned the
UEof
or colours, beliigcailed the black, blue, yellow, red, was selected each word of unknown. Ttel&itiaJ syllable
the symbol for the uaS&ys^.
Thus yd meant
a?;
M (from
;
Vi5VlS.w The Hindus
i^ere the Srst
atoolatelv negative numbers difference
between
f
and
to.
recognize the
and of irrational numbers. The numbers was brought out by
of "assets," to the other that at&acdxing to the one the idea * indicate opposite directions. them or of debts." by letting
was the recognition of two great step beyond Diophanlus 50 Bhaskara gives x Thus answers for quadratic equations. " 2 the he x 250. 45 ? But," says o for the roots of x ox
A
=


value
is
=
in this case not to be taken, for
it is
inadequate ;
A HISTORY OF MATHEMATICS
102
people do not approve of negative roots." Thus negative roots were seen, but not admitted. In the Hindu mode of solving quadratic equations, as found in Brahmagupta and Aryabhatta, it is believed that Greek processes are discernible. For
example, in their works, as also in Heron of Alexandria, + bx = o is solved by a rule yielding
ox2
This rule the
is
improved by Qridhara, who begins by multiplying of the equation, not by a as did his predecessors,
members
but by 4 a, whereby the possibility of fractions under the
He
radical sign is excluded.
gets
The most important advance in the theory of equations made in India three cases GKC* f
bx
=
c,
is
bx
affected quadratic the unifying under one rule of the
+ c = ore
2 ,
oo2
+ c=bx.
In Diophantus these cases seem to have been dealt with sepa he was less accustomed to deal with negative
rately, because 1
quantities.
An
advance far beyond the
Brahmagupta
is
Greeks and
even beyond
the statement of Bhaskara that " the square
of a positive, as also of a negative number, is positive that the square root of a positive number is twofold, positive and There is no square root of a negative number, for negative. ;
it is
not a square."
We have seen that the Greeks sharply discriminated between numbers and magnitudes, that the X
CANTOK, L,
irrational p. 625.
was not recognized
ABAB3
103
by them as a number. The discovery of the existence of was one of their profoundest achievements. To the Hindus this distinction between the rational and irrational irrationals
did not occur
was not heeded. They passed unmindful of the deep gulf separating the continuous from the discontinuous. Irrationals were sub jected to the same processes as ordinary numbers and were from one
;
at
any
rate, it
to the other,
indeed regarded by them as numbers. greatly aided the progress of mathematics results
arrived
processes would (p.
at
intuitively,
call for
much
By ;
so
doing they
for they accepted
which by
severely greater effort. Says
logical
Hankel
195 ), "if one understands by algebra the application of
arithmetical operations to complex magnitudes of all sorts, whether rational or irrational numbers or spacemagnitudes,
then the learned Brahmins of Hindostan are the real inventors of algebra."
In Bhaskara we find two remarkable
identities,
one of which
given in nearly all our school algebras, as showing root of a " binomial surd." find the
is
What
square
how
to
Euclid in
Book X. embodied hension,
is
in abstract language, difficult of compre here expressed to the eye in algebraic form and
applied to numbers
* :
Arabs
The Arabs
present an extraordinary spectacle in the history of civilization. Unknown, ignorant, and disunited tribes of
the Arabian peninsula, untrained in government and war, are p. 194.
A HISTOEY OF MATHEMATICS
104
in the course of ten years
fused by the furnace blast of
religious enthusiasm into a powerful nation, which, in one century, extended its dominions from India across northern
A hundred
years after this grand march of them assume the leadership of intellectual the Moslems became the great scholars of their
Africa to Spain. conquest, pursuits
;
we
see
time.
About 150 years
after
Mohammed's
flight
from Mecca
to
Medina, the study of Hindu science was taken up at Bagdad in the court of Caliph Almansur. In 773 A.D. there appeared at his court a Hindu astronomer with astronomical tables which
by royal order were translated into Arabic. These tables, known by the Arabs as the Sindhind, and probably taken from the Siddhdnta of Brahmagupta, stood in great authority. tables probably came the Hindu numerals. Except
With these
intercourse between
we possess no other evidence of Hindu and Arabic scholars yet we should
not be surprised
future historical research should reveal
for the travels of Albiruni,
if
;
Better informed are we as to the Arabic greater intimacy. of Greek Elsewhere we shall speak of learning. acquisition
geometry and trigonometry. Abtfl Wafd (940998) translated the treatise on algebra by Diophantus, one of the last of the Greek authors to be brought out in Arabic. Euclid, Apol
and Ptolemaeus had been acquired by the Arabic schol Bagdad nearly two centuries earlier.
lonius,
ars at
Of first
all
Arabic arithmetics
in historical importance
known is
to us, first in time and
that of
Muhammed
ibn
Mfod
Alchwarizmi who lived during the reign of Caliph Al Marnun (813833). Like all Arabic mathematicians whom we shall
name, he was
first
of all an astronomer
;
with the Arabs as with
the Hindus, mathematical pursuits were secondary. Alchwarizmi's arithmetic was supposed to be lost, but in 1857 a
Latin translation of
it,
made probably by Athelard
of Bath,
AEABS
105
in the library of the University of Cambridge. 1 arithmetic begins with the words, " Spoken has Algoritmi.
was found
The Let
us give deserved praise to God, our leader and defender." Here the name of the author, Alchwarizmt, has passed into ALgortimi,
whence comes our modern word algorithm) signifying the art of Alchwarizmi is familiar computing in any particular way. with the principle of local value and Hindu processes of calcu
lation. all
others in brevity and easiness, and exhibits the Hindu intel and sagacity in the grandest inventions." 2 Both in addition
lect
and
subtraction
Alchwarizmi proceeds from
left to right,
subtraction, strange to say, he fails to explain the case
a larger
digit is to be taken
from a smaller.
His
but in
when
multiplication
one of the Hindu processes, modified for working on paper is written over the corresponding digit in the multiplicand digits are not erased as with the Hindus, but
is
:
each partial product ;
The process
are crossed out.
The
idea.
136 24
of division rests on the
same
divisor is written below the dividend, the quotient
above it. The changes in the dividend resulting from the subtraction of the partial products are written
14g
above the quotient. For every new step in the divito the right one place. The the divisor moved is sion, author gives a lengthy description of the process in
46468
case of 46468
110
^
140
524
1 It
~ 324 = 143ff
, which is represented by This proc by the adjoining model solution. ess of division was used almost exclusively by early European writers who followed Arabic models, and it
Cantor
5
was found by Prince B. BOHCOMPAGNI under the title " Algoritmi He published it in his book Trattati d' ArU
de numero Indorum." metica,
Rome,
1857.
2
CANTOE, L, 713 8 CANTOE, I., 717. under Pacioli.
;
HANTLEL, p. 256. This process of division will be explained more fully
A HISTORY OF MATHEMATICS
106
1 Later Europe in the eighteenth century. Arabic authors modified Alchwarizmi's process and thereby Alchwarizmi approached nearer to those now prevalent.
was not extinct
in
explains in detail the use of sexagesimal fractions. Arabic arithmetics usually explained, besides the four cardi
9V
nal operations, the process of "casting out the (called sometimes the "Hindu proof"), the rule of "false position," the rule of " double position," square and cube root, and frac tions (written without the fractional line, as
The Rule
by the Hindus).
of Three occurs in Arabic works, sometimes in their It is a
remarkable fact that among the early Arabs no trace whatever of the use of the abacus can be discovered. algebras.
At the
close of the thirteenth century, for the first time,
we
an Arabic writer, Ibn Albannd, who uses processes which are a mixture of abaca! and Hindu computation. Ibn Albanna"
find
lived in Bugia,
an African
seaport,
and
it is
plain that he came
under European influences and thence got a knowledge of the abacus. 2 It is noticeable that in course of time, both in arithmetic
and algebra, the Arabs in the East departed further and further from Hindu teachings and came more thoroughly under This is to be deplored, for on these subjects the Hindus had new ideas, and by rejecting them the Arabs barred against themselves the road of prog the influence of Greek science.
Thus, Al Karchl of Bagdad, who lived in the beginning of the eleventh century, wrote an arithmetic in which Hindu ress.
numerals are excluded
!
All numbers in the text are written
out fully in words. In other respects the work is modelled almost entirely after Greek patterns. Another prominent and 1
2
also
HANKEL, p. 258. Whether the Arabs knew the use
of the abacus or not is discussed
by H. WEISSENEOE^T, Einjvhrung der
durch
Gterbertj pp. 59,
jetzigen Zi/ern in
Europa
AEABS
107
able writer, Abtfl Wafd, in the second lialf of
tlie tentli
cen
tury, wrote an arithmetic in which Hindu numerals find no The question why Hindu numerals are ignored by place.
authors so eminent, at one time there
is
certainly a puzzle.
may have been
Cantor suggests that which one
rival schools, of
followed almost exclusively Greek mathematics, the other Indian. 1
The algebra of Alehwarizmi is the first work in which the word "algebra" occurs. The title of the treatise is aldschebr wahnukdbala. These two words mean " restoration and oppo
By
sition.''
" restoration
ative terms to
??
was meant the transposing of neg the other side of the equation; by "opposi
from both sides of the equation of like terms so that, after this operation, such terms appear only on that side of the equation on which they were in excess. Thus,
tion/' the discarding
2x =
&= +
+3xr
2x 6 passes by aldschebr into 5 2 ar 2 walmukabala into 6 x. When this, by Alehwarizmi's aldschebr icdlmukdbala was translated into
5x* 4
3 z2
;
6
= +
and
Latin, the Arabic title
was
word was the form of
retained, but the second
word remaining in gradually discarded, the Such is the origin of this word, as revealed by the Several popular etymologies of the study of manuscripts. first
algebra.
word, unsupported by manuscript evidence, used to be current. For instance, "algebra" was at one time derived from the
name
of the Arabic scholar, Dschdbir ibn Aflah, of Seville,
who was
called Geber
by the Latins.
But Geber lived two two centuries after
centuries after Alchwarizini, and therefore
2 appearance of the word. Alchwarizmi's algebra, like his arithmetic, contains nothing
the
*
2
first
CANTOE, L, 765. See CANTOE, I., 719, 721,
72*2
;
HANSEL,
p.
248
;
PELIX MULLER, His
torisdietymologische Stwlien vber Mctthematiscke Terminologie^ Berlin, 1887, pp. 9, 10.
A HISTORY OF MATHEMATICS
108 original.
tion
of
It explains the
linear
elementary operations and the solu Whence did the
and quadratic equations.
That he got it Hindus had
author receive his knowledge of algebra? solely
from Hindu sources
is
impossible, for the
no such rules as those of "restoration" and "opposition"; they were not in the habit of making all terms in an equa is done by "restoration." Alchwarizmfs somewhat those of Diophantus. But we can not conclude that our Arabic author drew entirely from Greek sources, for, unlike Diophantus, but like the Hindus, he recognized two roots to a quadratic and accepted irrational It would seem, therefore, that the aldschebr wedsolutions. mukdbala was neither purely Greek nor purely Hindu, but was a hybrid of the two, with the Greek element predominating. In one respect this and other Arabic algebras are inferior to both the Hindu and the Diophantine models: the Eastern Arabs use no symbols whatever. With respect to notation,
tion
positive
as
rules resemble
* algebras have been divided into three classes (1) Rhetorical in which no are used, everything being Algebras, symbols written out in words. Under this head belong Arabic works :
(excepting those of the later Western Arabs), the Greek works of lamblichus and Thymaridas, and the works of the early
and of Eegiomontanus. The equation ic2 j 10 x = 39 was indicated by Alchwarizmi as follows: "A square and ten of its roots are equal to thirtynine dirhem that is, if Italian writers
;
you add
to a square ten roots, then this together equals thirty
nine." (2) Syncopated Algebras, in which, as in the first class, every
thing is written out in words, except that the abbreviations are used for certain frequently recurring operations and ideas. Such, are the works of Diophantus, those of the later Western
,
pp. 302306.
109
AEABS
Arabs, and of the later European writers down to about the As middle of the seventeenth century (excepting Vieta's). in take we a from an illustration sentence which, Diophantus, for the sake of clearness,
we
shall use the
Hindu numerals
and, in place of the Greek symbols, the corresponding English If S., X., U., m. stand for "square," "num abbreviations. " " ber," unity," minus," then the solution of problem IIL, 7, in Diophantus, viz., to find three numbers whose sum is a " Let square and such that any pair is a square, is as follows us assume the sum of the three numbers to be equal to the :
1 square 1 S. 2 X. 1 U. ? the first and the second together 1 S. then the remainder 2 X. 1 U. will be the third number. Let
the second and the third be equal to 1 S. 1 U. m. 2 X., of which the root is IX. m. 1 U. Xow all three numbers are 1 S. 2 X. 1 U.
;
hence the
first will
be 4 X.
But
this
and
the second together were put equal to 1 S., hence the second will be 1 S. minus 4 X. Consequently, the first and third ^
together, 6 X. 1 U.
must be a square.
Let this number be
121 U., then the number becomes 20 U. Hence the first is SO TL, the second 320 U., the third 41 U., and they satisfy the conditions."
Symbolic Algebras, in which all forms and operations by a fully developed symbolism, as, for ex In this class may be reckoned Hindu 39. a^ 10a; ample, (3)
are represented
+
=
works as well as European since the middle of the seventeenth century.
From
this classification
due to Xesselmann the advanced
ground taken by the Hindus is brought to full view; also The Arabs, the step backward taken by the early Arabs. however, made substantial contributions to what we may call geometrical algebra. Xot only did they (Alehwarizmi, Al 1
In our notation the expressions given here are respectively, 3,
x2, a;2~4x, Gx+l.
A HISTORY OF MATHEMATICS
110
Karchi) give geometrical proofs, besides the arithmetical, for the solution of quadratic equations, but they (Al ilah&ni, Abu 'far Alchazin, Abu'l Dschud, 'Omar Alchaijami) dis covered a geometrical solution of cubic equations, which alge The roots were braically were still considered insolvable.
Dscha
constructed by intersecting conies. 1 Al Karchi was the first Arabic author to give and prove the theorems on the summation of the series:
13
+ 23 + 33 +
...
Allusion has been
+ = (1 + 2 + 71
made
S
...
+
2
7l)
to the fact that the
Western Arabs
While in the East the developed an algebraic symbolism. geometric treatment of algebra had become the fashion, in the
West
the Arabs elaborated arithmetic and algebra inde
pendently of geometry. Of interest to us is a work by Atkalsddl of Andalusia or of Granada, who died in 1486 or 1477.*
His book was entitled Raising of the Veil of the Science of The word "gubar" meant originally "dust," and Gitbdr. stands here for written arithmetic with numerals, in contrast to mental arithmetic.
In addition, subtraction, and multipli
cation, the result is written above the other
square root is indicated dschidr,
^/Jg =
meaning "root, ^=..
The
figures.
by ^, the initial letter of the word 77
particularly "square root."
The proportion 7 12 :
= 84
:
x was written
Thus, ^
.'.
84
4fco
.\12.\7, the symbol for the unknown being here probably " imagined to be the initial letter of the word dsckahala, not to
know."
Observe that the Arabic writing
is
from right to left
In algebra proper, the unknown was expressed by the words i
Consult CANTOR,
*
CANTOR,
I.,
I.,
810817.
721, 725, 781, 776778
HANSEL,
pp. 274280.
E0BOPE DURING THE MIDDLE AGES schai or dschidr, for
= ^oj, a? = o, + 63 in this way
x
111
which Alkalsadi uses the abbreviations 2 Thus, he writes Sic = 12 #
(J for equality.
63
if
J^
This symbolism probably began to be developed among the Western Arabs at least as early as the time of Ibn Albanna It is of interest, when we remember (born 1252 or 1257). that in the Latin translations made by Europeans this sym
bolism was imitated.
Europe during
The the
the
Middle Ages
swamps and forests of and from the Ural Mountains swept down upon
barbaric nations, which from the
]S"orth
Europe and destroyed the Roman Empire, were slower than the Mohammedans in acquiring the intellectual treasures and the civilization of antiquity. The first traces of mathematical knowledge in the Occident point to a Eoman origin. Introduction of
Eoman
Arithmetic.
After Boethius and
Cassiodorius, mathematical activity in Italy died out. About one century later, Isidorus (570636), bishop of Seville, in
an
Spain, wrote
modelled after the
encyclopaedia,
and of Cassiodorius. ruvium.
He
entitled
It
Origines.
was
Eoman
encyclopedias of Martinus Capella Part of it is taken up with the quad
gives definitions of technical terms
and
their
but does not describe the modes of computation
etymologies ; then in vogue.
He
numbers into odd and even, speaks of perfect and excessive numbers, etc., and finally bursts out in admiration of number, as follows: "Take away number
from
all things,
divides
and everything goes to destruction." *
After
Isidorus comes another century of complete darkness; then 1
CANTOE, L, 824.
A HJSTOBZ OF MATHEMATICS
112
appears the English monk, Bede
His works contain
treatises
the
Venerable
(672735).
on the Gomputus, or the computa
and on fingerreckoning. a fingersymbolism was then widely used for
It appears that
tion of Eastertime,
The was a problem which in those days greatly agitated the ehurck The desira bility of having at least one monk in each monastery who calculation.
correct determination of the time of Easter
could determine the date of religious festivals appears to have Deen the greatest incentive then existing toward the study of
"The computation of Eastertime," says Cantor, "the real central point of timecomputation, is founded by
arithmetic.
Bede as by Cassiodorius and
others,
upon the coincidence,
once every nineteen years, of solar and lunar time, and makes no immoderate demands upon the arithmetical knowledge of the pupil
who aims
to solve simply this problem."
*
It is not
surprising that Bede has but little to say about fractions. In one place he mentions the Boman duodecimal division into ounces.
The year in which Bede died is the year in which the next prominent thinker, Alcuin (735804) was born. Educated in Ireland, he afterwards, at the court of Charlemagne, directed the progress of education in the great Frankish Empire. In the schools founded by him at the monasteries were taught the psalms, writing, singing, computation (computus), and gram mar. As the determination of Easter could be of no particular
word computus probably refers We are ignorant of the modes of It is not probable that Alcuin was
interest or value to boys, the
to computation in general.
reckoning then employed. familiar with the abacus or the apices of Boethius. He be longed to that long list of scholars of the Middle Ages and of the Renaissance
who dragged
the theory of numbers into
,
L, 831.
EUKOPE DUBING THE MIDDLE AGES
113
theology. For instance, the number of beings created by God, created all things well, is six, because six is a perfect
who
number (being equal 8
is
to the
a defective number, since
sum its
of
its
for that reason, the second origin of
the number
which
8,
is
divisors 1, 2, 3)
divisors 1
+ 2 f 4 < 8,
;
but and,
mankind emanated from
the number of souls said to have
been in Xoah's Ark,
There
is
a collection of "Problems for Quickening the is certainly as old as 1000 A.D., and possibly
Mind*' which older.
Cantor
earlier,
and by
of the opinion that it was written much Alcuin. Among the arithmetical problems of is
this collection are the fountainproblems
which we have en
countered in Heron, in the Greek anthology, and among the Hindus. Problem No. 26 reads: dog chasing a rabbit, which has a start of 150 feet, jumps 9 feet every time the rab
A
jumps 7, To determine in how many leaps the dog over takes the rabbit, 150 is to be divided by 2. The 35th problem is as follows dying man wills that if his wife, being with bit
:
A
child, gives birth to a son, the son shall inherit f and the widow J of the property ; but if a daughter is born, she shall
How
inherit ^ and the widow ^ of the property. is the if a to both son and a be divided, born ? are daughter property
This problem is of interest because by its close resemblance to a Roman problem it unmistakably betrays its Eoman originHowever, its solution, given in the collection, is different from Some of the the [Etonian solution, and is quite erroneous.
problems are geometrical, others are merely puzzles, such as the one of the wolf, goat, and cabbagehead, which we shall
The collector of these problems evidently mention again. aimed to entertain and please his readers. It has been re marked that the proneness to propound jocular questions is truly AngloSaxon, and that Alcuin was particularly noted in Of interest is the title which the collection this respect.
A HISTORY OF MATHEMATICS
114
bears: "Problems for Quickening the Mind." Do not these words bear testimony to the fact that even in the darkness of
the Middle Ages the minddeveloping power of mathematics was recognized ? Plato's famous inscription over the entrance of his academy is frequently quoted; here we have the less
weighty, but significant testimony of a people hardly yet intellectual slumber.
awakened from
During the wars and confusion which followed the
fall of
the empire of Charlemagne, scientific pursuits were abandoned, but they were revived again in the tenth century, principally He was born in Gerbert. through the influence of one man, Aurillac in Auvergne, received a monastic education, and en gaged in study, chiefly of mathematics, in Spain. He became
bishop at Hheims, then at Ravenna, and finally "was made Pope under the name of Sylvester II. He died in 1003, after
a
life
involved in
many
political
and
ecclesiastical quarrels.
Gerbert made a careful study of the writings of Boethius, and Rule of Computation on published two arithmetical works, the Abacus,
Now
and
A
Small Book on
for the first time do
computation.
we
the Division
of Numbers.
get some insight into methods of
Gerbert used the abacus, which was probably In his younger days Gerbert taught
unknown
to Alcuin.
school at
Eheims
the trivium and quadruvium being the and one of his pupils tells us that
subjects of instruction
Gerbert ordered from his shieldmaker a leathern calculating board, which was divided into 27 columns, and that counters of horn were prepared, upon which the first nine numerals (apices) were marked. Bernelinus, a pupil of Gerbert, de scribes the abacus as consisting of a
smooth board upon which
geometricians were accustomed to strew blue sand, and then to draw their diagrams. For arithmetical purposes the board
was divided tractions
into 30 columns, of which three were reserved for while the remaining 27 were divided into groups
EUROPE DURING THE MIDDLE AGES
Every group of the columns was
with three columns in each.
marked respectively by the and
S
115
letters
C (centum,
M (monas).
100),
D (decem,
Bernelinus gives the nine numerals used (the apices of Boethius), and then remarks that the Greek letters may be used in their place. 1 By the 10),
(singularis) or
use of these columns any number can be written without intro ducing the zero, and all operations in arithmetic can be per Indeed, the processes of addition, subtraction, and multiplication, employed by the abacists, agreed substantially with those of today. The adjoining figure shows the multi
formed.
plication of 4600
by
23.
2
The process
is
as
follows: 36 = 18; 3 x4 = 12; 26 = 12; 2x4 = 8; 1 + 2+2 = 5; remove the 1, 2, 1 + 1 + 8 = 10 re 2, and put down 5 ;
;
move
and put down 1 in the column
1, 1, 8,
next to the
left.
Hence, the sum 105800.
If counters were used, then our crossing
out of digits (for example, of the digits 1, 2y 2 in the fourth column) must be im agined to represent the removal of the counters
2
1, 2, 2,
3
and
the putting of a counter marked 5 in their place. If the numbers were written on sand, then the numbers 1, 2, 2 were erased and 5 written instead.
The
process of division
So
was entirely
different
from the
has this operation appeared that the of a quotient may almost be said to be foreign to concept
modern.
antiquity.
difficult
Gerbert gave rules for division
which apparently
were framed to satisfy the following three conditions (1) The use of the multiplication table shall be restricted as far as possible; at least, it shall never be required to multiply mentally a number of two digits by another of one digit; :
CJUSTTOK, I., 881.
8
FROBDUEIK, p. 106.
116
A HISTOEY OF MATHEMATICS
(2) Subtractions
shall be
avoided as much, as possible and
The operation shall proceed in a mechanical without That it way, purely requiring trials. should be necessary to make such conditions, will perhaps not seem so strange, if we recollect that monks of the Middle
replaced by additions; (3)
1
Ages did not attend school during childhood and learn the memory was fresh. Gerbert's
multiplication table while the
rules for division are the oldest extant.
They
are so brief as
be very obscure to the uninitiated, but were probably intended to aid the memory by calling to mind the successive to
steps of the process.
In later manuscripts they are stated
We illustrate this division by the 2 example 4087 = 6 = 681. The process is a kind of "complementary division." Begin nings of this mode of procedure are found more
fully.
the Eomans, but so far as known it was never used by the Hindus or Arabs. It is called "complementary" because, in our or 4 example, for instance, not 6, but is the number operated with. The rationale
among
106
of the process may, perhaps, be seen partial this
explanation
4000
:
f
10
below as part of the quotient.
too large a divisor
4.400
= 1600.
;
from
= 400,
this
write
But 10 is add
to rectify the error,
Then 1000^10 =
100,
write
below as part of the quotient to rectify this new error, add 4.100 = 400. Then 600 fthis
400
;
= 1000.
It will be
Divide
1000^10, and
so
on.
observed, that in
division, like this, it
complementary was not necessary to
p. 323. 2
Quoted by PRIEDLEIN, p. 109. The mechanism of the division is as follows: Write down the dividend 4087 and above it the divisor 6.
EUROPE DURING THE MIDDLE AGES snow
117
To a modern computer it would seem as though the above process of divi sion were about as complicated as human ingenuity could the multiplication table above the
make
5's.
1
Xo wonder
that it was said of Gerbert that he gave which were hardly understood by the most painstaking abacists; no wonder that the Arabic method of it.
rules for division
when first introduced into Europe, was called the " golden division (divisio aurea), but the one on the abacus " the iron division " (divisio ferrea). division,
"
The question has been asked, whence did Gerbert get Ms abacus and Ms complementary division? The abacus was probably derived from the works of Boethius, but the comple
mentary division
nowhere found
in its developed form before mainly his invention ? From one appears that he had studied a paper on multipli is
the time of Gerbert.
Was
it
of his letters it cation and division by " Joseph Sapiens," but modern research 2 has as yet revealed nothing regarding tMs man or his writings.
Above the 6
which
write: 4,
is
the difference between 10 and
6.
Multiply
this difference 4 into 4 in the
column I and move the product 16 to the erase the 4 in column I and write it in column C,
right by one column below the lower horizontal ;
line, as part of the quotient. Multiply the 1 write the product 4 in column C erase the 1 and write it below, onejeolumn to the right. Add the numbers in C, 6 + 4 10, and write 1 in I. Then proceed as before : 1.4 4, write it in C, and write 1 below. 4.4 16 in C and X, 4 below in 1.4 4 in 1 ;
in I
by
4,
;
=
= = 18 in C and X
= X
= X, below; = 4 in X, 1 below; 4 + 8 = 12 in C and X 1.4 = 4 in X, 1 below 2 + 4 = 6 in X, 6.4 = 24 in X and 1, 6 below 2.4 = 8 in I, 2 below 8 f 4 + 7 = 19 in X and I 1.4 = 4 in 1, 1 below 9 f 4 = 13 in X and I 1.4 = 4 in I, 1 below 3 + 4 = 7. Dividing 7 by 4
f
6
+
8
;
;
1.4
;
;
;
;
;
6 goes
and leaves
;
;
Write the 1 in I above and also below. Add the digits in the columns below and the sum 681 is the answer sought, i.&, 4087 ~ 6 = 681, leaving the remainder 1. 1
1
For additional examples of complementary division see FEIEDLEIN, GUNTHER, Math. Unterr. im, d. Mtttela^ pp* 102106, Consult H. WEISSENBOEN, Einfuhrung der jetzigen Ziffern in
pp. 109124 2
1.
^
;
Berlin, 1892.
A HISTORY OF MATHEMATICS
118
In course of the next
the instruments for
five centuries
abacal computation were considerably modified. Xot only did the computing tables, strewn with sand, disappear, but also Gerbert's abacus with vertical columns and
marked counters
In their place there was used a calculating board (apices). with lines drawn horizontally (from left to right) and with all alike and unmarked. Its use is explained in the printed arithmetics, and will be described under Eecorde.
counters first
The new instrument was employed 1 England, but not in Italy. Translation of Arabic Manuscripts.
which were made
in
Germany, Prance,
Among
the transla
the period beginning with the twelfth century is the arithmetic of Alchwarizmi (probably translated by Athelard of Bath), the algebra of Alchwarizmi tions
in
(by Gerard of Cremona in Lombardy) and the astronomy of Al Battanl (by Plato of Tivoli). John of Seville wrote a liber
by him from Arabic authors. Thus Ara and algebra acquired a foothold in Europe. Arabic or rather Hindu methods of computation, with the aJghoarismi, compiled
bic arithmetic
zero and the principle of local value, began to displace the abacal modes of computation. But the victory of the new
over the old was not immediate. The struggle between the two schools of arithmeticians, the old abacistic school and the
new
algoristic school,
was incredibly
long.
The works
issued
by the two schools possess most striking differences, from which it would seem clear that the two parties drew from in dependent sources, and yet it is argued by some that Gerbert got his apices and his arithmetical knowledge, not from Boethius, but from the Arabs in Spain, and that part or the whole of the geometry of Boethius is a forgery, dating from the time of Gerbert. If this were the case, then we should i
See CAJTTOB, Vol. EL, 2nd Edition, 1900, pp. 216, 217 ; see also GERGetchichte ckr Mathevnatik in Deutschland, Miincheu, 1877, p. 29.
HAKBT,
EUROPE DUBISG THE KIDDLE AGES
119
expect the writings of Gerbert to betray Arabic sources, as da those of John of Seville. But no points of resemblance are found.
Gerbert could not have learned from the Arabs the
use of the abacus, because we possess no reliable evidence that the Arabs ever used it The contrast between algorists and abacists consists in this, that unlike the latter, the former
mention the Hindus, use the term algorism, calculate with the The former teach the zero, and do not employ the abacus. extraction of roots, the abacists do not; the algorists teach
sexagesimal fractions used by the Arabs, while the abacists employ the duodecimals of the Eomans.
Towards the close of the twelfth The Firsi Awakening. century there arose in Italy a man of genuine mathematical power. He was not a monk, like Bede, Alcuin, and Gerbert, but a business man, whose leisure hours were given to mathe
To Leonardo of Pisa, also called Fibonacci^ or we owe the first renaissance of mathematics on
matical study. Fibonaciy
When
a boy, Leonardo was taught the use In later years, during his extensive travels in Egypt, Syria, Greece, and Sicily, he became familiar with Christian
soil.
of the abacus.
various
modes
of
computation.
Of the
several processes
After his he found the Hindu unquestionably the best return home, he published in 1202 a Latin work, the liber abaci.
A
second edition appeared in 1828.
While
this
book
contains pretty much the entire arithmetical and algebraical knowledge of the Arabs, it demonstrates its author to be more
than a mere compiler or slavish imitator. The liber abaci begins thus "The nine figures of the Hindus are 9, 8, 7, 6, 5, :
4, 3, 2, 1.
With
which in Arabic
and with this sign, 0, any number may be written."
these nine figures
is
called sifr,
= empty) passed into the Latin zephirum sifr (stfra and the English cipher. If it be remembered that the Arabs wrote from right to left, it becomes evident how Leonardo^ The Arabic
A HISTORY OF MATHEMATICS
120
in the above quotation,
came
to write the digits in descend
ing rather than ascending order; it came to write 4182 instead of 182.
plain also how he liber abaci is the
is
The
a recurring series. 1 Interest ing is the following problem of the seven old women, because it is given (in somewhat different form) by Annies, 3000 years earliest
earlier
work known
to contain
Seven old women go to Rome, each woman has seven
:
mules, each mule carries seven sacks, each sack contains seven loaves, with each loaf are seven knives, each knife rests in
seven sheaths. 137256. 2
What
is
sum
the
total of all
named ?
Ans.
Leonardo's treatise was for centuries the storehouse
from which authors drew material for their arithmetical and Leonardo's algebra was purely "rhetor algebraical books. "
that is, devoid of all algebraic symbolism. Leonardo's fame spread over Italy, and Emperor Frederick The presentation II. of Hohenstaufen desired to meet him.
ical
,
was John of
of the celebrated algebraist to the great patron of learning
accompanied by a famous
scientific
tournament.
Palermo, an imperial notary, proposed several problems which Leonardo solved promptly. The first was to find the number x,
such that
answer
is
a^
+ 5 and 3? o are each square numbers. The 3& for (3^) + 5 = (&)* (3^)  5 = (2^)
x=
2
2
2
.
;
;
The Arabs had already solved
similar problems, but
some parts
The second The + jgeneral equations was unknown at that
of Leonardo's solution seem original with him.
problem was the solution of algebraic solution of cubic
10 a;
2 x*
3?
= 20.
time, but Leonardo succeeded in approximating to one of the
He
roots.
gave x
1^27"4
f
"3Jf
4T40ti
r
,
the answer being
thus expressed in sexagesimal fractions. Converted into deci mals, this value furnishes figures correct to nine places. These
and other problems solved by Leonardo i
CANTOS, EL, 26.
2
disclose brilliant
CANTOB,
II., 26.
EUEOPE DUKING THE MIDDLE AGES talents.
His geometrical
writings
will
121
touched upon
be
later.
In Italy the Hindu numerals were readily accepted by the enlightened masses, but at first rejected by the learned circles. Italian merchants used them as early as the thirteenth cen tury; in 1299 the Florentine merchants were forbidden the use of the Hindu numerals in bookkeeping, and ordered either to use the Koman numerals or to write numbers out in words. 1
The reason for this decree lies probably in the fact that the Hindu numerals as then employed had not yet assumed fixed, definite shapes,
and the variety
of forms for certain
digits
sometimes gave rise to ambiguity, misunderstanding, and fraud. In our own time, even, sums of money are always written out in words in case of checks or notes.
Among
the Italians are
evidences of an early maturity of arithmetic. Says Peacock, "The Tuscans generally, and the Florentines in particular,
whose
city
was the cradle of the
literature
and
arts of the thir^
teenth and fourteenth centuries, were celebrated for their know ledge of arithmetic ; the method of bookkeeping, which is called especially Italian, was invented by them; and the operations of arithmetic, which were so necessary to the proper conduct of their extensive commerce, appear to have been cultivated
and improved by them with particular care; indebted
.
.
.
to
them we arc
for the formal introduction into books of arith
metic, under distinct heads, of questions in the single
and
double rule of three, loss and gain, fellowship, exchange, sim ple interest, discount,
compound
interest,
and so on."*
In Germany, France, and England, the Hindu numerals were scarcely used, until after the middle of the fifteenth 8 A small book on Hindu arithmetic, entitled De arte century. numerandi, called also Mgorismusy was read, mainly in France i
HAJTKBL, p. 341.
*
PEJLCOCK, p. 414,
* See
UHGBK,
p. 14.
A HISTORY OF MATHEMATICS
122 and
Italy, for several centuries.
It
is
usually ascribed to John
Halifax, also called Sacrobosco, or Hofywood, who was born in Yorkshire, and educated at Oxford, but who afterwards set
and taught there until his death, in 1256^ The booklet contains rules without proofs and without numerical tled in Paris
examples;
many
it
times
was printed in 1488 and
It
ignores fractions. later.
De Morgan
According to
it is
the
first
work ever printed in a French town (Strasbourg)." 1 Here and there some of our modern notions were anticipated
arithmetical
by writers of the Middle Ages.
For example, Nicole Oresme, a bishop in Xormandy (about 13231382), first conceived the notion of fractional powers, afterwards rediscovered by Stevin,
and suggested a notation
V4 =
8, it
2 3 Thus, since 4 = 64 and In Oresme's notation 4 1* is
for them.
follows that 4
1*
= 8.
Such suggestions to the con
expressed,
trary notwithstanding, the fact remains that the fourteenth and fifteenth centuries brought forth comparatively little in the
way
of original mathematical research.
writers, but their scientific efforts
There were numerous
were vitiated by the methods
ofjcliolasttc thinking.
GEOMETRY
ASTD
TEIGO^OMETBY
Hindus
Our account of Hindu geometrical research brief; for, in the first place, like the
will be very
Egyptians and Romans,
the Hindus never possessed a science of geometry; in the second place, unlike the Egyptians and Romans, they do not 1
See Siblioth.
Mathem., 1894, pp. 7378;
CJLNTOR, IL, pp, 8891
4t ;
De
arte
HALLIWELL, in Sara Mathematica, *CANTOB, II., 133.
J. 0.
also 1895, pp.
uumerandi" was reprinted 1839.
3637; last
bj
HIM>US
123
figure in geometrical matters as the teachers of other nations,
Their
came
was original, their later geometry and China. Brahmagnpta gives the
early geometry
from
Greece
"Heronic Formula" for the area of a
triangle.
He
also
gives the proposition of Ptolemaeus, that the product of the diagonals of a quadrilateral is equal to the sum of the product
of the opposite sides, and adds new theorems of his own on The calculation of areas quadrilaterals inscribed in a circle.
forms the chief part of Hindu geometry.
^^ywcrf Bhaskara' s
Interesting
proof
of
Aryabhatta gives
is
the
theorem of the right tri He draws a right angle. triangle four times in the
square of the hypotenuse, so that in the middle there remains a square whose side equals the difference between the two sides of the right triangle. Arranging this small square and the four triangles in a different way, they are seen, together, to make up the sum of the squares of the two sides. "Be
hold," says Bhaskara, without adding another word of ex Bigid forms of demonstration are unusual with planation.
the Hindus. writer.
This proof
is
Pythagoras' proof
given
much
earlier
may have been
by a Chinese In this.
like
another place Bhaskara gives a second demonstration of this
theorem by drawing from the vertex of the right angle a perpendicular to the hypotenuse, and then suitably manipulat ing the proportions yielded by the similar triangles. This proof
was unknown
Europe until it was rediscovered by Wallis. More successful were the Hindus in the cultivation of in
As with the Greeks, so with them, it was valued merely as a tool in astronomical research. Like the Babylonians and Greeks, they divide the circle into 360 de
trigonometry.
grees and 21,600 minutes.
Taking =ai416, and 2 ^=21,600,
A HISTOBT OF MATHEMATICS
124 they got r
= 3438
that
;
is,
the radius contained nearly 3438
This step was not Grecian. Some Greek mathematicians would have had scruples about measur
of these circular parts.
ing a straight line by a part of a curve. With Ptolemy the division of the radius into sexagesimal parts was independent of the
no common unit of The Hindus divided each quadrant
division of the circumference
measure was
selected.
;
into 24 equal parts, so that each of these parts contained 225
A
out of the 21,600 units. vital feature of Hindu trigonometry that they did not, like the Greeka, reckon with the whole
is
chord of double a given arc, but with the sine of the arc (i.e. half the chord of double the arc) and with the versed sine of
The
the arc.
entire chord
AB was
called
by the Brahmins ,/y$ or jiva,' words which meant also the cord of a hunter's bow. ^
D
For AC, or half the chord, they used the words jy&rdha or ardhajyd, but the names
B
of the whole chord were also used for It is interesting to trace the his
brevity.
tory of these words.
The Arabs
transliterated jivd or jtva into
they afterwards used the word dschaib, of " the same form, meaning bosom." This, in turn, was nearly dschiba.
For
this
Thus arose translated into Latin, as sinus, by Plato of Tivoli. the word sine in trigonometry. For "versed sine," the Hindus Observe used the term ufkramajyd, for "cosine," Jcotijyd. 1 the three of Hindus used our that trigonometric functions, while the Greeks considered only the chord.
The Hindus computed a
table of sines
by a
theoretically
simple method. The sine of 90 was equal to the radius, or of 60 was also 3438, therefore 3438; the chord of an arc
AB
half this chord
AC,
or the sine of 30, was 1719.
,
I.,
658, 736.
Applying the
125
ARABS formula
sin* a f cos*
a ^r2 and observing that ,
they obtained sin 45 equal sin (90
a),
= A/^ = 2431.
and making a
sin
45= cos 45,
Substituting for cos a
its
= 60, they obtained
the sines of 90, 60, 45 as startingpoints, they reckoned 2 the sines of half the angles by the formula versin 2 a =2 sin a,
With
thus obtaining the sines of 22 30', 15, 11 15', 7 30', 3 45'. They now figured out the sines of the complements of these 67 30'; angles, namely, the sines of 86 15', 8230 , 78 45', 75, then they calculated the sines of half these angles; thereof f
and so on. By this very simple process 1 of all the angles at intervals of 345'. the sines they got No Indian treatise on the trigonometry of the triangle is
their complements,
In astronomical works, there are given solutions of and spherical right triangles. Scalene triangles were
extant.
plane divided
up
into right triangles,
tations could be carried out.
whereby
As the
all
ordinary compu
table of sines gave the
values for angles at intervals of 3 degrees, the sines of inter vening angles had to be found by interpolation. Astronomical observations
and computations possessed only a passable
1 degree of accuracy.
Arabs
The Arabs added hardly anything
to the ancient stock of
Yet they play an allimportant rdle in mathematical history; they were the custodians of Greek and Oriental science, which, in due time, they transmitted to geometrical knowledge.
i
A. ABSOSTH, Die Gesekickte der reinen Mathematik, Stuttgart, 1852, This work we shall cite as ABJTBTH. See, also,
pp. 172, 173. p, 217. 3
AKSBTH,
p. 174.
A HISTOB? OF MATHEMATICS
126 the
The startingpoint for all geometric study the Arabs was the Elements of Euclid. Over and over
Occident.
among
again was this great work translated by them into the Arabic tongue. Imagine the difficulties encountered in such a trans
Here we see a people,
just emerged from barbarism, untrained in mathematical thinking, and with limited facilities Where was to be found for the accurate study of languages. lation.
the man, who, without the aid of grammars and dictionaries, had become versed in both Greek and Arabic, and was at the same time a mathematician? How could highly refined scientific thought be conveyed to undeveloped minds by an
not strange that sev eral successive efforts at translation had to be made, each
undeveloped language translator resting
?
Certainly
it is
upon the shoulders of
his predecessor.
In Arabic rulers wisely enlisted the aid of Greek scholars. and were the medicine, sciences, especially philosophy Syria
by Greek Christians. Celebrated were the schools and Emesa, and the Nestorian school at Edessa. After the sack and ruin of Alexandria, in 640, they became the
cultivated
at Antioch
chief repositories in the East of Greek learning.
Elements were translated into Syriac. Christians were called to Bagdad, the
Euclid's
Prom Syria Greek Mohammedan capital.
During the time of the Caliph Hdritn arRaschld (786809) was
made
the
first
Arabic translation of Ptolemy's Almagest; also (first six books) by Haddschddsck ibn
of Euclid's Elements 1 Jilsuf ibn Matar.
He made
a second translation under the
Caliph Al
Mamun
dition, in
a treaty of peace with the emperor in Constanti
(813833).
This caliph secured as a con
CANTOR, I., 702 Biblioth. Mathem., 1892, p. 65. An account of trans commentators on Euclid was given by Ibn All Ja'kfib anNacfim In his Fihrist, an important bibliographical work published in Arabic in 987. See a German translation by H. SUTEE, in the Zettschr. fur Math. u. 1
;
lators and
2,
Supplement, pp. 387.
127
ARABS
number of Greek manuscripts, which he ordered Euclid's Elements and the Sphere translated into Arabic. Ja'ktib and Cylinder of Archimedes were translated by Ishdk ibn Hunain, under the supervision of his father Hunadn nople, a large
AM
lators,
At
1
These renderings were unsatisfactory ; the trans though good philologians, were poor mathematicians.
ibn Ishdk.
this time there
were added to the thirteen books of the
Elements the fourteenth, by Hypsicles (?), and the fifteenth by Damascms (?}. It remained for Tdbit ibn Kurrah (836901) to bring forth an Arabic Euclid satisfying every need. Among other important translations into Arabic were the mathemati
works of Apollonius, Archimedes, Heron, and Diophantus. Thus, in course of one century, the Arabs gained access to the
cal
vast treasures of Greek science.
A
later
and important Arabic edition of Euclid's Elements gifted JVosir Eddin (12011274), a Persian
was that of the
astronomer who persuaded his patron Hul,gu to build him and He tried his his associates a large observatory at Maraga. skill at
In
a proof of the parallelpostulate.
all
such attempts,
some new assumption is made which is equivalent to the thing Thus Nasir Eddin assumes that if AB is _L to to be proved. CD at C, and if another straight line EDF makes the angle
EDO
acute, then the
perpendiculars to
AB, comprehended
CD toward E, ! from It is are further <7D. the and are shorter shorter, they unless can be this in case how difficult to see otherwise, any between
AB and EF, and drawn on the
side of
Nasir sne looks with the eyes of Lobatehewsky or Bolyai " had some influence on the later " Eddin's proof development His edition of Euclid was. printed of the theory of parallels. " was " in Arabic at Borne in 1594 and his brought ant proof *
CAXTOK, L, 703.
2
The prx>f
Mathem.,
is
given by KASTNER,
1892, p. 5.
I.,
375381 and in part, in
A HISTOEY OF MATHEMATICS
128
In Latin translation by Wallis in 1651. 1 Of interest is a new proof of the Pythagorean Theorem which Xasir Eddin adds to
the Euclidean proof. 2 An earlier demonstration, for the special case of an isosceles right triangle, 3 is given by 3fuhammed ibn
M&sd
Alchicarizml who lived during the reign of Caliph Al Mamtin, in the early part of the ninth century. Alchwarizmfs meagre treatment of geometry, as contained in his work on It Algebra, is the earliest Arabic effort in this science.
Hindu influences. Besides the Hindu values IT = VT5 and Arabic works, Hindu geometry hardly
bears unmistakable evidence of
value
TT
=
3f, it
*= iooJSJS
^
n
ever shows itself
contains also the later
Greek geometry held undisputed sway. In a book by the sons of M&S& ibn Schdkir (who in his youth was a robber) is given the Heronic Formula for the area of a tri angle.
;
A neat piece of research is displayed in the " geometric (940998), a native of Buzshan the improved theory of draughting by show to construct the corners of the regular polyedrons on
constructions" by Abtfl in Chorassan.
Wafd
He
ing how the circumscribed sphere. Here, for the first time, appears the condition which afterwards became very famous in the Occident, that the construction be effected with a single open ing of the compasses.
The
best original
work done by the Arabs in mathematics
1
WALLIS, Opera, II., 669673. See H. SUTER, in Biblioth. 3fathem., 1892, pp. 3 and 4. In Hoffmann's and "Wipper's collections of proofs for this theorem, Nasir Eddtn's proof is given without any reference to him. 8 Some Arabic writers, Beh Eddin for instance, called the Pythagorean 2
Theorem the "figure
of the bride."
This romantic appellation originated
probably in a mistranslation of the Greek word wJ/t^i?, applied to the theorem by a Byzantine writer of the thirteenth century. This Greek word admits of two meanings, " bride" and " winged insect." The fig ure of a right triangle with its three squares suggests an insect, but Beha
Eddin apparently translated the word as " bride." in ISIntermediaire des Mathematiciens, 1894, T.
I.,
See PAUL p. 264.
AKABS is
129
the geometric solution of cubic equations aad the develop of trigonometry. As early as 773 Caliph Almansur came
ment
into possession of the
Hindu
table of sines, probably taken
from Brahmagupta's Siddhdnta. The Arabs called the table the Sindhind and held it in high authority. They also came into early possession of Ptolemy's Almagest and of other Greek astronomical works. Muhammed ibn Mtisd Akhivarizml was
engaged by Caliph Al
Mamun
in
making
extracts
from the
Sindhindj in revising the tables of Ptolemy, in taking obser vations at
Bagdad and Damascus, and in measuring the degree Remarkable is the
of the earth's meridian. derivation,
by Arabic authors, of formulae
in spherical trigonometry, not from the "rule of six quantities of Menelaus/' as u rule of four previously, but from the
PP
QQ
be l l and quantities." This is: If two arcs of great circles intersecting in Ay
and if PQ and drawn of be arcs circles perpendicular to QQ^ then PiQi great we have the proportion sin^UP:
sinPQ =
sin APi
:
sinP^.
This departure from the timehonoured procedure adopted by Ptolemy was formerly attributed to Dschdbir idn Aflah, bat recent study of Arabic manuscripts indicates that the transi " to the "rule of four tion from the "rule of six quantities 1 quantities" was possibly effected already by Tdbto ibn Knrrak
(836901), the change being adopted by other writers who preceded DschSbir ibn Aflak* Foremost among the astronomers of the ninth century ranked *
2
H. SuTxm, in Biblioth. Maikem., 1893, p. 7. For the mode of deriving formulae for spherical
rigiit triangtes,
ac
cording to Ptolemy, also according to I>scl>abir ibn Allah, and nis Arabic predecessors, see HJLKKEL, pp. 285287 ; CANTOS, L, 794.
A HISTORY OF MATHEMATICS
130
Al Battdnl, called Albategnius by the Latins. Battan in Syria was his birthplace. His work, De scientia stellarum, was translated
century.
into Latin
by Plato Tiburtinus, in the twelfth In this translation the Arabic word dschiba, from
the Sanskrit jtva,
is
said to have been rendered by the word
sinus; hence the origin of "sine." Though a diligent student of Ptolemy. Al Batt^ni did not follow him altogether. He
took an important step for the better, when he introduced the Indian sine " or half the chord, in place of the whole chord of Ptolemy.
Another improvement on Greek trigonometry
made by the Arabs
points
likewise to Indian
influences:
Operations and propositions treated by the Greeks geometri Thus Al cally, are expressed by the Arabs algebraically.
Batt&ni at once gets from an equation
cos0 a process unknown to
by means of sin = D f VI + i* To the formulae known Greek antiquity. 2
,
1
= D, the value of
^5
to
Ptolemy he adds
an important one of his own for obliqueangled spherical angles namely, cos a = cos b cos c f sin b sin c cos A.
tri
;
Important are the researches of Abtfl Wafd.
He
invented
a method for computing tables of sines which gives the sine of half a degree correct to nine decimal places. 2 He and AlBatttoi, in the study of shadowtriangles cast by sundials, introduced the trigonometric functions tangent and cotangent; he used also the secant and cosecant. An important change in method was inaugurated by Dschdbir ibn Aflah of Seville in Spain (in the second half of the eleventh century) and by
Nas\r Eddln (12011274) in distant Persia. In the works of the last two authors we find for the first time trigonometry developed as a part of pure mathematics, independently of astronomy. *
CANTOR,
I., p,
7S7.
*
Consult CJLNTOE,
I,,
746748.
EUROPE DURING THE MIDDLE AGES
131
We
do not desire to go into greater detail, but would em phasize the fact that Naslr Eddln in the far East, during a
temporary cessation of military conquests by Tartar rulers, developed both plane and spherical trigonometry to a very 1 remarkable degree. Suter enthusiastically asks, what would
have remained for European scholars of the fifteenth century to do in trigonometry, had they known of these researches ?
Or were some
To
of them, perhaps,
this question
we
aware of these investigations ? no final answer.
can, as yet, give
Europe during
the
Middle Ages
Before the introduction Introduction of Roman Geometry. of Arabic learning into Europe, the knowledge of geometry in the Occident cannot be said to have exceeded that of the
Egyptians in 600
The monks
B.C.
of the Middle
Ages did not
go much beyond the definitions of the triangle, quadrangle, circle, pyramid, and cone (as given in the Roman encyclo pedia of the Carthaginian, Martianus Capella), and the simple In Alcutn's " Problems for Quickening rules of mensuration. the Mind," the areas of triangular and quadrangular pieces of land are found by the same formulae of approximation as
those used by the Egyptians and given by Boethius in his geometry: The rectangle equals the product of half the sums
the triangle equals the product of half of two sides and half the third side. After Alcuin ?
of the opposite sides
sum
the
;
the great mathematical light of Europe was Gerbert (died the geometry of Boethius and 1003). In Mantua he found
studied self
i
it zealously.
It is usually believed that
was the author of a geometry.
For further
pp. 18.
particulars consult
Ms
Gerbert him
This contains nothing
article in JB&lioth.
Matkem^
1893,
A HISTORY OP MATHEMATICS
132 more
tlian the
sional errors in
geometry of Boethius, but the fact that occa BoetMus are herein corrected shows that the
A
author had mastered the sub 1 " The earliest
mathemat
ject.
ical
a
paper of the Middle Ages
which deserves letter of
this
name
is
a
Gerbert to Adalbold,
2 bishop of Utrecht," in which is explained the reason why
J/
the area of a triangle, obtained " " geometrically by taking the altitude, differs from the area calcu
product of the base by its lated " arithmetically," according to the formula Ja(afl), used by surveyors, where a stands for a side of an equilateral triangle.
latter is
Gerbert gives the correct explanation, that in the all the small rectangles, into which the triangle
formula
supposed to be divided, are counted in wholly, even though
parts of them project beyond it. Translation of Arabic Manuscripts.
The beginning of
the
twelfth century was one of great intellectual unrest. Philos ophers longed to know more of Aristotle than could be learned
through the writings of BoetMus; mathematicians craved a profounder mathematical knowledge. Greek texts were not at hand; so the Europeans turned to the Mohammedans for instruction; at that time the
Arabs were the great scholars
We
read of an English monk, Aihelard of Bath, who travelled extensively in Asia Minor, Egypt, and Spain, braving a thousand perils, that he might acquire the of the world.
language and science of the Mohammedans. 1
For description of
its
contents see CANTOR, I, 861867 ; S. im deutcben MiUelalter, Ber
Greschichte des SfatkematisGken Unterrickts lin,
*
1887, pp. 115120.
ELuntEL, p. 314.
"The Moorish
EUROPE BTJEIKG THE MIDDLE AGES universities of
Cordova and Seville and Granada were dan
gerous resorts for Christians." 1
tables of
lation
in
He
1120.
Muhammed Euclid
of
suspicion that translation.
He made what
is
probably the
from the Arabic into Latin of Euclid's
earliest translation
Elements,
138
ibn
translated
also
the
Mus, AlehwarizmL
from the
Athelard was
Arabic aided
astronomical
In his trans
there
is ground for a by previous Latin
2
All important Greek mathematical works were translated from the Arabic. Gerard of Cremona in Lombardy went to
Toledo and there in 1175 translated the Almagest. We are told that he translated into Latin TO works, embracing the 15
books of Euclid, Euclid's Data, the Sphverka of Theodosius, and a work of Menelaus. A new translation of Euclid's 1
We
are surprised to read in the Dictionary of National Biography [Leslie Stephen's] that "it has not yet been determined whether the translation of Euclid's Elements . . . was made from the Arabic version
or from the original." To our knowledge no mathematical historian now donbts that the translation was made from the Arabic or suspects that Athelard used the Greek text. See CAIHX>B, I., 713, 906 II., 100. HANKEL, "VV. W. p. 335; S. GUSTTHER, Math. Unt. im d. Mffielalt.,pp. 147149 R. BALL, 1893, p. 170; Gow, p. 206; H. SUTEB, Gesch. d Math., 1., ;
;
146 ; HOEFER, Histoire des Mathematiques, 1879,
p. 321.
Remarkable
is
the fact that MAEIE, in his 12volume history of mathematics, not even He says that Campanus "a donn6 des Elements mentions Athelard. d'Euclide la premiere traduction qu'on ait p. 168. *
ette
en Europe."
MARTS,
II,,
In a geometrical inannscript in the British said that geometry was invented in Egypt by Eocleides.
CAICTOK, II., 100102.
Museum
it is
This verse
is
appended:
" Thys
craft
Yn tyme
com ynto England,
of
as
y ghow
say,
good Kyng Adelstones day."
See HALLIWBLL'S Eara Mathematica, London, 1841, p. 66, etc. As King Athelstan lived about 200 years before Athelard, it would seem that a Latin Euclid (perhaps only Hie fragments given by Boethius) "was known In England long before Athelard.
A HISTORY OF MATHEMATICS
134
Elements was made, about 1260, by Giovanni Campano
(latin
Campanus) of JSTovara in Italy. It displaced the ones and formed the basis of the printed editions.
ized form, earlier
The
The First Awakening.
central figure in mathematical
history of this period is the gifted
Leonardo of Pisa (1175?).
His main researches are in algebra, but his Practica Geometries, published in 1220, is a work disclosing skill and geometric
The writings of Euclid and of some other Greek known to him, either directly from Arabic manu or from the translations made by his countrymen, scripts Gerard of Cremona and Plato of Tivoli. Leonardo gives ele gant demonstrations of the "Heronic Formula" and of the
rigour.
masters were
theorem that the medians of a triangle meet in a point (known He also gives the to Archimedes, but not proved by him). theorem that the square of the diagonal of a rectangular parallelepiped is equal to the sum of the three squares of its 1 sides. Algebraically are solved problems like this: To in scribe in
an equilateral triangle a square resting upon the base
of the triangle.
A
geometrical work similar to Leonardo's in Italy was brought out in Germany about the same time by the monk
Jordanus Nemorarius.
It
was entitled De
printed by Curtze in 1887.
triangulis,
and was
It indicates a decided departure
from Greek models, though to Euclid reference is frequently is nothing to show that it was used anywhere as a textbook in schools. This work, like Leonardo's, was probably made. There
read only by the
we
As specimens
elite.
give the following
:
of remarkable theorems,
If circles can be inscribed
and circum
scribed about an irregular polygon, then their centres do not coincide ; of all inscribed triangles having a common base, the isosceles is the
tdon of
maximum.
Jordanus accomplishes the
trisec
an angle by giving a graduated ruler simultaneously ,
II., p. 39.
a
EUKOPE DUBIHG THE MIDDLE AGES
135
rotating and a sliding motion, its final position being fixed with aid of a certain length marked on the ruler. 1 In this trisection he does not pemit himself to be limited to Euclid's postulates,
which allow the use simply of an unmarked ruler
and a pair of compasses. He also introduces motion of parts of a figure after the manner of some Arabic authors. Such motion, The same mode of trisection is foreign to Euclid's practice.*
was given by Campanus. Jordanus's attempted exact quadrature of the circle lowers in our estimation. Circlesquaring now began to com
"him
mand
the lively attention of mathematicians.
Their efforts
futile as though they had attempted to jump into a rainbow; the moment they thought they had touched the goal, it vanished as by magic, and was as far as ever from
remained as
In their excitement many of them became sub and imagined that they had actually attained their aim, and were in the midst of a triumphal arch of glory, the wonder and admiration of the world.
their reach.
ject to mental illusions,
The
fourteenth and fifteenth centuries have brought forth
no geometricians who equalled Leonardo of Pisa, Much was written on mathematics, and an effort put forth to digest the rich material acquired from the Arabs. tributions were made to geometry.
No
substantial con
An
English manuscript of the fourteenth century, on sur veying, bears the title: Nowe sues here a Tretfe of Geometri
wherby you may kn&we the heghte, depnes, and the Bmfe ofmostwhed erthdy ihynges* The oldest French geometrical manu Like the Eng script (of about 1275) is likewise anonymous. lish treatise, it deals with mensuration. 1
From
the study of
CJLNTOE, IL, 81, gives the construction in full. fuller extracts from the De triangutts, see CANTOS, IL, S. GUNTHEE, op, cU., 160162. * See HJLLLIWEIX, Sara Mathematica, 5671 ; CAKTOB, II., p. 112, 2
For
A
J36
HISTORY OF MATHEMATICS
manuscripts, made to the present time, it would seem thai since the thirteenth century surveying in Europe had de
parted from
Roman models and come
completely under the An author of consid
influence of the GrsecoArabic writers. 1
was Thomas Bradwardine (1290 ?1349), He was educated at Merton Col lege, Oxford, and later lectured in that university on theology, philosophy, and mathematics. His philosophic writings con tain able discussions of the infinite and the infinitesimal sub in studied came to be connection which thenceforth with jects mathematics. Bradwardine wrote several mathematical treat erable prominence
archbishop of Canterbury.
A
was printed in Paris in 1511 as the work of Bradwardinus, but has been attributed by some to ises.
a Dane,
Greometria speculative*,
named
Petrus, then a resident of Paris.
ble
A
work enjoyed a wide
This
remark
popularity.
It
treats of the regular solids, of isoperimetrical figures in the
starpolygons.
manner of Zenodorus, and of The first appearance of such
polygons was with Pythagoras and his school. The pentagramstar was used by the Pythag oreans as a badge or symbol of recognition, and was called 2 next meet such polygons in the geom by them Health.
We
etry of
Boethius, in the translation of
Euclid
from the
Arabic by Athelard of Bath, and by Campanus, and in the earliest French geometric treatise, mentioned above. Brad
wardine develops some geometric properties of starpolygons their construction
and anglesum.
We encounter
these fas
cinating figures again in Eegiomontanus, Kepler, and others. In Bradwardine and a few other British scholars England
proudly claims the earliest European writers on trigonome Their writings contain trigonometry drawn from Arabic try. *
CANTOR, H.,
127.
a
Gow,
p. 161.
EUROPE DURING THE MIDDLE
AGES
137
John Maudith, professor at Oxford about 1340, umbra ("tangent"); Bradwardine uses the
sources.
of the
speaks terms umbra
recta (" cotangent ") and umbra versa (" tangent ")* which occur in Gerard of Cremona's translation of AlArzakeFs Toledian tables. The Hindus had introduced the sine, versed
the Arabs the tangent, cotangent, secant, cosecant Perhaps the greatest result of the introduction of Arabic What was learning was the establishment of universities.
sine, cosine;
their attitude toward
mathematics?
At the University
of
In 1336 a rule was intro
Paris geometry was neglected.
duced that no student should take a degree without attending lectures on mathematics, and from a commentary on the first six books of Euclid, dated 1536, it appears that candidates
for the degree of
A.M. had
to
2 attended lectures on these books.
take oath that they had Examinations, when held
at all, probably did not extend the nickname "
beyond the first book, as is " magister matheseos applied to the theorem of Pythagoras, the last of the book. At Prague,
shown by
founded in 1384, astronomy and applied mathematics were
Roger Bacon, writing near the close of the thirteenth century, says that at Oxford there were few additional requirements.
who
cared to go beyond the first three or four propo and that on this account the fifth proposition was called "eiefuga," that is, flight of the wretched." students
sitions of Euclid,
We
" are told that this fifth proposition was later called the pons " or " the 8 Clavius in his Euclid, asinorum Bridge of Asses." edition of 1591, says of this theorem, that beginners find it 1
CANTOS,
II., 111.
HANEJSL, pp. 354359. We have consulted also H. SUTEE, Die Mathomatik auf den Unwersitaten des MittelaUers, Zurich, 1887 ; S. GATHER, 2
Math. Unt. im d. Mitiela^ p. 199 CANTOR, II., pp. 189142. & This nickname is sometimes also given to the Pythagorean Theorem, L, 47, though usually I., 47, is called "the windmill." Bead THOMAS 1 CAMPBELL'S poem, "The Pons Asinorum.' ;
A
138
fflSTOBY
OF MATHEMATICS
and
and obscure, on account of tlie multitude of lines These angles, to which they are not yet accustomed.
last
words, no doubt, indicate the reason
difficult
why
geometrical
study seems to have been so pitifully barren. Students with no kind of mathematical training, perhaps unable to perform the simplest arithmetical computations, began to memorize the abstract definitions and propositions of Euclid. Poor absence an of with and preparation poor teaching, combined rigorous requirements for degrees, probably explain this flight In the middle of the fif this "elefuga."
from geometry
teenth century the
first
two books were read
at Oxford.
Thus it is seen that the study of mathematics was main tained at the universities only in a halfhearted manner.
;;^6
CITY,
MODERN TIMES
AEITHMETIC Jte
Development as a Science and Art
the sixteenth century the human mind made an extraordinary effort to achieve its freedom from scholastic and
DURING
ecclesiastical bondage.
This independent and vigorous intel mathematical books of the
lectual activity is reflected in the
work of the fifteenth as also emanated from Italian writers, Lucas Pacioli and Tartaglia. Lucas Pacioli (1445 ?1514 ?) also called Lnca di Burgo, Luca PociwoZo, or Pacciuolus was a Tuscan monk who taught mathematics at Perugia, His treatise, Naples, Milan, Florence, Home, and Venice.
time.
The
best
arithmetical
of the sixteenth century
Summa
de Arithmetica, 14&4, contains all the knowledge of
day on arithmetic, algebra, and trigonometry , and is the first comprehensive work which appeared after the The earliest arithmetic known to have liber abaci of Fibonaci. his
been brought out in print anywhere appeared anonymously in " Treviso, Italy, in 1478, and is known as the Treviso arithmetic." Tartaglia's real name was Nieoto Fontana (1499 ?1557). a boy of six, Nicolo was so badly cut by a French
When
he never again gained the free use of his tongue. Hence he was called Tartaglia, i.e. the stammerer. His widowed mother being too poor to pay his tuition at school, he soldier, that
139
A HISTOEY OF MATHEMATICS
140
learned to read and acquired a knowledge of Latin, Greek,
and mathematics without a teacher. Possessing a mind of extraordinary power, he was able to teach mathematics at an
He
early age. It
Brescia,
taught at Verona,
was his intention
to
Piacenza,
embody
Venice,
his
searches in a great work, General trattato di numeri
but at his death
was
it
still
unfinished.
The
and
original re et
misure,
two parts
first
were published in 1556, and treat of arithmetic. Tartaglia discusses commercial arithmetic somewhat after the manner of Pacioli, but with greater fulness and with simpler and more methodical treatment. His work contains a large num
ber of exercises and problems so arranged as to insure the reader's mastery of one subject before proceeding to the next. Tartaglia bears constantly in mind the needs of the practical man. His description of numerical operations embraces seven different modes of multiplication and three methods of divis 1 He gives the Venetian weights and measures. Mathematical study was fostered in Germany at the close of the fifteenth century by Georg Purbach and his pupil,
ion.
Hegiomontanus.
The first printed German arithmetic appeared
in 1482 at Bamberg.
It is by Ulneh Wagner, a practitioner of Kilrnberg. It was printed on parchment, but only fragments of one copy are now extant. 2 In 1483 the same Bamberg
publishers brought out a second arithmetic, printed on paper,
and covering 77 pages.
Wagner
is
believed to be
The work its
is
anonymous, but
XJlrich
worthy of remark arithmetic appeared only four
author.
It is
that the earliest printed German years after the first printed Italian arithmetic.
The Bam
berg arithmetic of 1483, says Unger, bears no resemblance to previous Latin treatises, but is purely commercial. Modelled after
it is
the arithmetic by John
.
60.
Widmann, *
UKGEB, pp.
Leipzig, 1489.
3640.
MODERN TIMES
141
This work has become famous, as being the earliest book in + and have been found. They occur in connection with problems worked by " false position." Wid
which the symbols
mann
"what is that is minus; what is +, that is The words "minus" and "more," or "plus," occur long before Widmann's time in the works of Leonardo of Pisa, who uses them in connection with the method of says,
,
more."
" false position in the sense of " positive error l " " error."
While Leonardo uses
minus
also
and " negative to indicate an
" operation (of subtraction), he does not so use the word plus." " The word Thus, 7 j 4 is written septem et quatuor." "plus," signifying the operation of addition, was first found
by Enestrom in an Italian algebra of the fourteenth century. The words " plus " and " minus," or their equivalents in tha modern tongues, were used by Pacioli, Chuquet, and Widmann. It is proved that the sign j comes from the Latin e, as it was cursively written at the time of the invention of printing. The origin of the sign is still uncertain. These signs were
used in Italy by Leonardo da Yinci very soon after their appear in. Widmann's work. They were employed by Grammateus
ance
(Heinrich Schreiber), a teacher at the University of Vienna, by Christoff Eudolff in his algebra, 1525, and by Stifel in 1553.
Thus, by slow degrees, their adoption became universal. During the early half of the sixteenth century some of the most prominent Grerman mathematieians (Grammateus, Rudolif, Apian, Stifel) contributed toward the preparation of
1
D
Intermediairc des McttMmatidens, 18&4, pp. 119consult also EKEsxadai Regarding the supposed origin of f and in Ofversigt af Eongl. Vetenskaps AJcademiens F&rhandlingar, Stock holm, 1894, pp. 243256 CANTOR, II., 211212 j DE MOEGAX in F$&&sophical Magazine^ 20, 1842, pp. 135137 ; in Trans, of the FbOo*.
G. ExESTBoac in
120.
;
Soc. of Cambridge, 11, p. 203 10,
p/182.
;
Enestr5m, BiUioth*
Jfotft. (3) 9, p.
155;
A HISTORY OF MATHEMATICS
142
arithmetics, but
after that period this important 1 The most the hands of the practitioners alone. the textbook Adam writers was of Riese, who early popular several 1522 of is the one arithmetics, but that published practical
work
fell into
usually associated with his name. French work which in point of merit ranks with Pacioli's Summa de Arithmetic but which was never printed before the
A
nineteenth century, is Le Triparty en la science des nombres, written in 1484 in Lyons by Nicolas Chuquet? contempo
A
rary of Chuquet, in France,
was Jacques
Lefevre,
who brought
out printed editions of older mathematical works. For instance, in 1496 there appeared in print the arithmetic of the German
monk, Jordanus Nemorarius, a work modelled after the arith metic of Boethius, and at this time over two centuries old.
A
quarter of a century later, in 1520, appeared a popular arithmetic by Estienne de la Roche, named also
French
The author draws
Villefranche.
We topics.
his
material
mainly from
Pacioli. 8
Chuquet and
proceed
Down
now
to the
discussion of a few arithmetical
to the seventeenth century great diversity
and
clumsiness prevailed in the numeration of large numbers. Italian authors grouped digits into periods of six, others
sometimes into periods of three.
Adam
Eiese,
who
did more
than any one else in the first half of the sixteenth century toward spreading a knowledge of arithmetic in Germany, writes 86
325
78
178,
and reads,
Sechs
und achtzig
tausend, tausend mal tausend, sieben hundert tausend mal tausendt, neun vnnd achtzig tausend mal tausend, drei huntausent,
fiinff:
vnnd zwantzig tausend,
ein hundert, acht
E, p. 44. 2
The Triparty
A description of 8
is
the
CJLNTOK, EL, 371.
printed in Bulletin Boncompagni, XIII., $85592, is given by CANTOR, IL, 347355.
work
MODERN TIMES und and
143
1 Stif el in 1544 writes 2 329 089.562 800, siebentzig." " duo millia millies millies millies trecenta ; viginti reads,
novem
millia millies millies
;
oetoginta
novem
millia millies
;
quingenta sexaginta duo millia; octingenta." Tonstall, in 9 2 This habit of 1522, calls 10 "millies millena millia."
grouping digits, for purposes of numeration, did not exist among the Hindus. They had a distinct name for each suc step in the
cessive
scale,
and
has been remarked that
it
this fact probably helped to suggest to
them the
principle of
They read 86789325178 as follows: "8 kharva, 6 padma, 7 yyarbuda, 8 koti, 9 prayuta, 3 laksha, 2 ayuta, 5 sahasra, 1 Qata, 7 daQon, 8." 3 One great objection to this
local value.
Hindu scheme
is
that
it
burdens the memory with too
many
names.
The
first
improvement on ancient and mediaeval methods was the invention of the word miUione by the
of numeration
Italians in the fourteenth century, to signify great thousand*
This new word seems originally to have indicated a concrete measure, 10 barrels of gold. 5 The words millione, or 1000
2
.
nulla or cero (zero) occur for the first time in print in the Arithmetica of Borgi (1484). In the next two centuries the use of millione spread to other European countries. Tonstall, in 1522, speaks of the term as common in England, but rejects
barbarous! The seventh place in numeration he calls " millena 7 millia; vulgus millionem barbare vocat." Diicange
it as
of
Eymer mentions the word
million in 1514
8 ;
in 1540
it
occurs
of the
words
once in the arithmetic of Christof? Rudolff.
The next decided advance was the introduction
" in " 1 WILDERMTTTH, article Rechnen Encyklopcedie des gesammten Erziehungs und UnterricJitswesens^ DR. K. A. SCHMID, 1885, p. 794. 2 8 *
PEACOCK, p. 426. HANKEL, p. 15. PEACOCK, p. 378.
5
HANKEL,
6
CANTOR,
p. 14. II.
,
305.
7 8
PEACOCK, p. 426. WILDERMTJTH.
A HISTOKY OF MATHEMATICS
144
Their origin dates back almost to tlie first used. So far as known,
billion, trillion, etc.
time when the word million was
occur in a manuscript work on arithmetic by that French physician of Lyons, Nicolas Chuquet He em gifted ploys the words byllion, tryllion, quadrillion, quyllion, sixlion,
they
first
octyllion, nonyllion, "et ainsi des aultres se plus on voulait proceder/' to denote the second, third, etc. 1 3 Evi powers of a million, i.e. (1000,000/, (1000 OOQ) etc.
septyUion,
oultre
5
,
dently Chuquet had solved the difficult question of numera The new words used by him appear in 1520 in the tion. printed work of La Roche. Thus the great honour of having simplified numeration of large numbers appears to belong to the French. In England and Germany the new nomenclat ure was not introduced until about a century and a half later. In England the words billion, trillion etc., were new when Locke wrote, about 1687. 2 In Germany these new terms appear for the first time in 1681 in a work by Heckenberg of Hanover, but they did not come into general use before the ,
3
eighteenth century. About the middle of the seventeenth century it became the custom in France to divide numbers into periods of three digits, instead
of six, and to assign to the
word
billion,
in
2 12 place of the old meaning, (1000,000) or 10 , the new meaning 9 4 of 10 * The words trillion, quadrillion, etc., receive the new
definitions of
1012, 1C 15,
billion, trillion, etc.,
countries, 1
2 8 *
etc.
mean
At the present time the words
in France, in other southEuropean
and in the United States
(since the first quar
CANTOR, IL, 348. LOCKE, Human Understanding, Chap. XVI. UNGER, p. 71. Dictionnaire de la
Langue Franqaise par E. LITTRE.
It is interest
ing to notice that Bishop Berkeley, when a youth of twentythree, pub lished in Latin an arithmetic (1707), giving the words billion, trillion, etc.
,
with their
new meanings.
MODERN TIMES
etc. while in Germany, Eng and other northEuropean countries they mean 10 12,
ter of this century),
land,
10 18,
10 9 10 12
145
,
,
;
etc.
From what has been
said
it
appears that, while the Arabic
notation of integral numbers was brought to perfection by the Hindus as early as the ninth century, our present numeration One of the dates from the close of the fifteenth century.
advantages of the Arabic notation is its independence of its numeration. At present the numeration, though practically adequate, is not fully developed. To read a number of, say,
1000
digits,
or to read the value
decimal places by William Shanks, new words.
A
of
v,
calculated to 707
we should have
to invent
good numeration, accompanied by a good notation, is work in numbers. We are told that
essential for proficient
the Yancos on the
Amazon
could not get beyond the number
three, because they could not express that idea by any phrase 1 ology more simple than Poettarrarorincoaroac.
As
regards arithmetical operations, it is of interest to notice Hindu custom was introduced into Europe, of begin an addition or subtraction, sometimes from the right, ning but more commonly from the left. Notwithstanding the in that the
convenience of this latter procedure, it is found in Europe 2 as late as the end of the sixteenth century.
Like the Hindus, the Italians used
many
different
methods
in the multiplication of numbers. An extraordinary passion seems to have existed among Italian practitioners of arith metic, at this time, for inventing
new
forms.
8 Tartaglia speak slightingly of these efforts.
Pacioli
and
Pacioli himself
gives eight methods and illustrates the first, named bericuocoli or scacherii, by the first example given here : 4 1 2
PEACOCK, PEACOCK,
p. 390.
3
p. 427.
*
PEACOCK, PEACOCK,
p. 431.
p.
429; CANTOR,
IT.,
311.
A HISTOKY OF MATHEMATICS
146
9876 6789
9876 6789 61101000
5431200 475230 40734 67048164
7i9iQQj8
[6191'S2
67048164
The second method, for some unknown reason called castel i.e. "by the little castle," is shown here in the second
lua'o,
Th^j third method (not illustrated here) invokes example. the aid of tables; the fourth, crocetta sine casetta, i.e.
"by
cross multiplication," though harder than the
practised by the Hindus (named by
was them the "lightning" others,
See our third method), and greatly admired by Pacioli. example, In which the product is built up as follows :
3
6 f 10 (3 1 f 5 6) + 100 (3 4 + 5 1 + 1000 (54 + 2.1) + 10,000 (2 4). .


+
2

6)
.
In
Pacioli's fifth method, quadrttatero, or "
by the square," the
digits are entered in a square divided like a chessboard in
are
added diagonally.
gelosia
105248
or graticola,
tiplication"
987 x 987). lattice
Hindu
(see
fashion,
His sixth i.e.
our
is
"latticed
fourth
up and
called
mul
example,
named because the figure looks like a such as was then placed in Venetian
It is so
or grating,
windows that ladies and nuns might not easily be seen from the street. 1
The word jealousy.
gelosia
The
methods are
last
means primarily two of Pacioli's
illustrated, respectively,
by the examples 234x48=234x6x8 and 163 x 17 = 163 x 10 + 163 x 7. 1
PEACOCK, p. 431.
MODEEN TIMES
147
Some books also
give the mode of multiplication devised by the imitation of one of the Hindu methods. We in Arabs early It would be illustrate it in the example .359 x 837 == 300483. l
erroneous to conclude from the existence of these and other
methods that of fact the
of
all
them were
in actual use.
of Pacioli's methods (now in
first
As a matter common use)
was the one then practised almost exclusively. The second method of Pacioli, illustrated in our
^ ^
0798 9672
second example, would have been a better choice, as
we expect It is
to
show later. remark that the Hindus and
worthy of
3 S
Arabs apparently did not possess a multiplication table, such, for instance, as the one given 2 by Boethius, which was arranged in a We show it here as far as 4 x 4. The
4x4,
books.
sometimes
Some
occurs
writers
(for
in
35999 355 3
square. Italians
Another gave this table in their arithmetics. form of it, the triangular, shown here as far as
1234 2468 3 6
sembling
a
exceeding
5.
first
process
9 12
arithmetical
instance,
Pinseus
l
2
in Prance and Hecorde in England) teach also a kind of complementary multiplication, re
Romans. It was intended
2 7 3
7
4
found among the
frequently called the "sluggard's rule," and to relieve the memory of all products of digits is
3
It is analogous to the process in Gerbert s
comple
" mentary division : Subtract each digit from 10, and write down the differences together, and add as many tens to their
product as the
X Q
4 difference."
3
(first 
If a
i first
and b designate the
rule rests on the identity (10 p. 77.
I digit exceeds the
(second)
*
a)(10
digits,
&)+10(o+&
BOETHIUS (FfclEDLEIN'S Ed.), p. 432.
>
then the
10)=o&. p. 53.
A HISTORY OF MATHEMATICS
148
Division was always considered an operation of considerable Pacioli gives four methods. The first, "division difficulty. the by head," is used when the divisor consists of one or digit
two
(such as 12, 13), included in the Italian tables of multiplication. 1 In the second, division is performed The third successively by the simple factors of the divisor. digits
method, "by giving," is so called because after each subtrac tion we give or add one more This figure on the right hand. is
the method of " long division "
expended
now prevalent. But Pacioli his enthusiasm on the fourth method, called by the
Italians the
"galley," because the digits in the completed in the form of that vessel. He considered
work were arranged
this procedure the swiftest, just as the galley
was the
The English call it the scratch method. division of 59078 by 74 is shown in Fig. I. 2
swiftest
The complete
ship.
62
6
W
t 19 59078(
zp
in
$078(7
#
74
7
yn Fie.
1.
FIG.
The other four
2.
figures
FIG. 3.
show the
FIG.
jr
FIG.
4.
5.
division in its successive
8
stages. 1
PEACOCK, p. 432. This example is taken from the early German mathematician, Purbach, by ARXO SADOWSKI, Die osterreichiche Rechenmethode, KGnigsberg, Pacioli's illustration of the galley method is 1892, p. 14. given by PEACOCK, p. 433, UNGEB, p. 79. 8 The work proceeds as follows : Tig. 2 shows the dividend and divisor 2
written in their proper positions, also a curve to indicate the place for the quotient Write 7 in the quotient ; then 7 x 7 49 = 10 ; write 49, 59 10 above, scratch the 59 and also the 7 in the divisor. 7 x 4 = 28 the 4 ;
=

MODERN TIMES For a long time the
galley or scratch
149
method was used almost
to the entire exclusion of the other methods.
It
in
was adopted the works
in
As
late as the
was preferred to the one now in vogue. Spain, Germany, and England. It is found
seventeenth century of
it
Tonstall,
Napier, and Oughtred.
Kecorde, Stifel, Stevin, Wallis, until the beginning of the
Kot
1 It will eighteenth century was it superseded in England. be remembered that the scratch method did not spring into
existence in the form taught by the writers of the sixteenth On the contrary, it is simply the graphical repre century. sentation of the method employed by the Hindus, who calcu lated with a coarse pencil on a small dustcovered tablet. The erasing of a figure by the Hindus is here represented by the
scratching of a figure. On the Hindu tablet, our example, taken from Purbach, would have appeared, when completed, as follows
26
:
59078
798
74
The
practice of
"
operations by
European arithmeticians to prove their n casting out the 9's was another method, useful
to the Hindus, but poorly adapted for computation on paper or slate, since in this case the entire operation is exhibited at
the close, and
all
the steps can easily be reexamined.
being under the in the dividend, 28 must be subtracted from 100 the remainder is 72 scratch the 10, the of the dividend, and the 4 of the above write the 7 and 2. Write down the divisor one divisor (Fig. 3) ;
;
;
place further to the right, as in Fig. = 63; 7263 = 9; scratch 72
9x7 9x4 = 36;
97
36 = 61;
7 in the dividend
move the
7 into 72 goes 9 times 7 below, write 9 above
scratch the 9 above, write 6 above
and write 1 above it scratch 7 and 4 below,
;
;
scratch the 5 above
the dividend write 6, Fig. 1
Now
it
PEACOCK,
p. 434.
1.
and write 2
The remainder
;
;
;
;
scratch
Fig. 5.
divisor one place to the right. It goes 8 times ; 7 5 ; scratch 6 and 1 above and write 5 above 1 ; 8
 56 = 32 = 26 58
61
4.
and the
x8 x4
Again
= 56 = 32
; ;
above the scratched 8 in
is 26.
A HISTORY OF MATHEMATICS
150
As there were two rival methods of division ("by giving" and " galley "'), there were also two varieties in the extraction 1 In ease of surds, great interest was of square and cube root. Leonardo rules of approximation. of the in taken discovery and others give the Arabic rule (found, for the works of the Arabs, Ibn Albanna and Alkal
of Pisa, Tartaglia, instance, in sadi),
which may be expressed
in our algebraic symbols thus
2 :
This yields the root in excess, while the following Arabic rule makes it too small :
Similar formulae were devised for cube root.
In other methods of approximation to the roots of surds, the idea of decimal fractions makes its first appearance, though About the their true nature and importance were overlooked. middle of the twelfth century, John of Seville, presumably in imitation of Hindu methods, adds 2 n ciphers to the number, then finds the square root, and takes this as a numerator of a The fraction whose denominator is 1, followed by n ciphers. it failed to be gen Italian his even contemporaries for other by erally adopted, at least mentioned by been would have it wise certainly devoted in a work Cataldi (died 1626) exclusively to the ex
same method was followed by Cardan, but
;
Cataldi finds the square root by means of a method ingenious and novel, but for continued fractions traction of roots.
practical purposes inferior to Cardan's.
Orontius Einseus, in
France, and William Buckley (died about 1550), in England, 1
For an example
p. 436. * Consult
of square root
by the scratch method, see PEACOCK,
CANTOR, L, 808; PEACOCK,
p. 436.
MODBRK TIMES
151
In extracted the square root in the same way as Cardan. root of and the adds six Finseus finding 10, square ciphers, lOOOOOOOr 3
conc ^ U(^ es as shown here. The 3 162 expresses the square root in decimals. new branch of arithmetic decimal thus fractions
162
1
A
<50
9
720
60
stared
43 200
him
in the face,
his predecessors
12
tional part to 9'
43"
it
had many of But he !
thinking of sexagesimal and to reduce the frac hastens fractions, sees it not;
3
he
as
and contemporaries
sexagesimal
12"'.
mals in a case
is
What was needed like
this ?
an integer, 1 thus,
of
divisions
for the discovery of deci
Observation,
keen
observation.
And
yet certain philosophers would make us believe that observation is not needed or developed in mathematical
study! Close approaches to the discovery of decimals were made in other ways. The German Christoff Rudolff performed divis ions by 10, 100, 1000, etc., by cutting off by a comma ("mit 2 as many digits as there are zeros in the einer virgel ") divisor.
The honour Simon Stevin
of the invention of decimal fractions belongs to
of Bruges in Belgium (15481620), a man remarkable for his varied attainments in science, for his inde
pendence of thought, and extreme lack of respect for authority. It would be interesting to know exactly how he came upon his in Flemish (later in great discovery. In 1584 he published French) an interest 8
table.
" I hold
it
now
next to certain,"
"that the same convenience which has
De Morgan, always dictated the decimal form for tables of compound In interest was the origin of decimal fractions themselves/' says
1585 Stevin published his i
PEACOCK,
La Disme
p. 437. 8
Arithmetical Books,
(the fourth part of a 2
CANTOR,
p. 27.
H,
399.
A HISTORY OF MATHEMATICS
152
French work on mathematics), covering only seven pages, in which decimal fractions are explained. He recognized the full importance of decimal fractions, and applied them to all
No
the operations of ordinary arithmetic, fect at its birth.
invention
is
In place of our decimal point, he used a cipher
notation.
per
Stevin's decimal fractions lacked a suitable ;
to
each place in the fraction was attached the corresponding index. Thus, in his notation, the number 5.912 would be IS 3
5@9i
These indices, though cumbrous, are (i)2(). herein we shall find the principle of because interesting an exponen another important innovation made by Stevin As an illustration of Stevin's notation, we tial notation. 5912 or
append the following
844362
$(!><*<>
1
2
% ?
96
by
&
~
2 (
%
$ $
P
^ ^
%
?
$
2(3
(
1
He was enthusiastic, not only over decimal fractions, but also over the decimal division of weights and He considered it the measures.
(2>
e*>
<
division.
(
5 8 7
duty of governments to establish As to decimals, he latter.
the
says, that, while their introduction
%
man
may be delayed, "it is in the future remains the
certain
same as it is now, then he will not always neglect so great an advan His decimals met with ready, though not immediate, tage." His La Disme was translated into English in recognition. 1608 by Bichard Norton. A decimal arithmetic was published 2 in Ijondon, 1619, by Henry Lyte. As to weights and measures, little did Stevin suspect that two hundred years wouM elapse before the origin of the metric system and that at the close of tfee nineteeBiili century England and the New World wwld be hopelessly bound by the chains o custom to tihe use that if the nature of
;
* PKJLOOCBL,
p. 440.
MODERN TIMES
153
of yards, rods, and avoirdupois weights. But we still hope that the words of John Kersey may not be proved prophetic " It being improbable that such a Reformation will ever be brought :
to pass, I shall proceed in directing a Course to the Studious for obtaining the frugal Use of such decimal fractions as are
in his Powers."
1
After Stevin, decimals were used on the continent by Joost by birth, who prepared a manuscript on arith
Burgi, a Swiss
metic soon after 1592, and by Johann Hartmann Beyer, who assumes the invention as his own. In 1603 he published at Frankfurt on the Main a Logistica Decimalis. With Burgi, a zero placed underneath the digit in unit's place answers as a sign of separation. Beyer's notation resembles Stevin's, but it
may have
been suggested to him by the sexagesimal notation He writes 123.459872 thus
then prevalent.
:
o
i
n
in rv v vi
123
4
5
9
7
8

2.
VI
Again he writes .000054 thus, 54, and remarks that these differ from other. fractions in having the denominator written
The decimal point, says Peacock, is in 1617 published his Rabdologia con taining a treatise on decimals, wherein the decimal point is above the numerator.
due
to Napier,
who
In the English translation of executed Napier's Descriptio, by Edward Wright in 1616, and corrected by the author, the decimal point occurs on the first
used in one or two instances.
page of logarithmic
tables.
There
is
in English arithmetics between 1619
no mention of decimals and 1631. Oughtred, in
1631, designates .56 thus, 056. Albert Girard, a pupil of John Wallis, Stevin, in 1629 uses the point on one occasion. in 1657, writes 12 [345, but afterwards in his algebra adopts the KERSEY'S WINGATE, 16tb Ed., London, 1735, p. 119. quotes the same passage from the 2d Ed., 1638. 1
WILDERMUTH
A HISTORY OF MATHEMATICS
154
Georg Andreas Bockler,
usual point.
in
Ms
Arithmetica
Niirnberg, 1661, uses the comma in place of the point (as do the Germans at the present time), but applies decimals only to 1 De Morgan 2 the measurement of lengths, surfaces, and solids.
says that "it was long before the simple decimal point was fully recognized in all its uses, in
England
at least,
and on
the continent the writers were rather behind ours in this
As long
matter.
as
Oughtred was widely used, that
till
is,
the end of the seventeenth century, there must have been a large school of those who were trained to the notation 123 [456. first quarter of the eighteenth century, then, we must not only the complete and final victory of the decimal point, but also that of the now universal method of performing the operation, of division and extraction of the square root."
To the
refer,
The progress
of
the decimal
notation, and of all other, " The history of language
is
interesting and
is
of the highest order of interest, as well as utility
instructive.
.
its
;
.
.
sug
gestions are the best lesson for the future which a reflecting
mind can have." (De Morgan.) To many readers it will doubtless seem that after the Hindu notation was brought to perfection probably in the ninth century, decimal fractions should have arisen at once in the minds of mathematicians, as an obvious extension of it.
But 3 "
it is
how much science had attempted and how deeply numbers had been pon
curious to think
in physical research,
was perceived that the allpowerful simplicity of the Arabic Notation was as valuable and as manageable in an infinitely descending as in an infinitely ascending pro dered before
it
'
e
gression."
The experienced 1
WILDERMUTH.
8
NAPIEK, MAJRK.
1834, Chap.
H
teacher has again and again 2
made
Arithmetical Books, p. 26.
observa
Memoirs of John Napier of Merchiston, Edinburgh,
MODERN TIMES
155
on the development of thought, in watch the of his The mind of the pupil, like ing progress pupils. the mind of the investigator, is bent upon the attainment of some fixed end (the solution of a problem), and any con tions similar to this,
not
sideration
directly
involved
in
this
immediate
aim
Persons looking for some frequently escapes his vision. particular flower often fail to see other flowers, no matter how pretty. One of the objects of a successful mathematical teacher, as of a successful teacher in natural science, should
be to habituate students to keep a sharp lookout for other things, besides those primarily sought, and to make these, too, subjects of contemplation.
Such a course develops investi
gators, original workers.
The miraculous powers of modern calculation are due to the Hindu Notation, Decimal Fractions, and Logarithms. The invention of logarithms, in the first quarter of the seventeenth century, was admirably timed, for Kepler was three inventions
:
then examining planetary orbits, and Galileo had just turned the telescope to the stars. During the latter part of the fif teenth and during the sixteenth century, German mathema
had constructed trigonometrical tables of great accuracy, but this greater precision enormously increased the work of the It is no exaggeration to say with Laplace that the calculator.
ticians
invention of logarithms
"by
the life of the astronomer."
shortening the labours doubled Logarithms were invented by
John Napier, Baron of Merchiston, in Scotland (15501617). of thirteen, Napier entered St. Salvator College, An uncle once wrote to Napier's father, "I St. Andrews.
At the age pray you,
Sir, to
send John to the schools either of France
no good at home." So he was In 1574 a beautiful castle was completed for him on the banks of the Endrick. On the opposite side of the river was a lint mill, and its clack greatly disturbed or Flanders, for he can learn
sent abroad.
A HISTOBY OF MATHEMATICS
156
He sometimes desired the miller to stop the mill Kapler. so that the train of his ideas might not be interrupted. 1 In 1608, at the death of his father, he took possession of Merchiston castle. 2 Napier was an ardent student of theology and astrology, and delighted to show that the pope was Antichrist. More worthy of his genius were his mathematical studies, which he
pursued as pastime for over forty years. Some of his mathe matical fragments were published for the first time in 1839.
The
great object of his mathematical studies was the simpli fying and systematizing of arithmetic, algebra, and trigo
nometry.
Students
analogies,"
and "Napier's
tion of spherical
in
trigonometry
right triangles.
happiest example of
remember
"Napier's
rule of circular parts," for the solu
artificial
This
memory
is,
that
perhaps, is
known."
"the In
1617 was published his Habdologia, containing "Napier's rods " or " bones " 3 and other devices designed to simplify This work was well known on multiplication and division. the continent, and for a time attracted even more attention than his logarithms. 1
As
late as
1721 E. Hatton, in his arith
Diet. Nat. Biog.
"A this connection the title of the following book is interesting. Bloody Almanack Foretelling many certaine predictions which shall come With a calculation concerning the to passe this present yeare 1647. time of the day of Judgment, drawne out and published by that famous astrologer, the Lord Napier of Marcheston." For this and for a catalogue of Napier's works, see MACDONALD'S Ed. of NAPIER'S Construction of the 2
In
Wonderful Canon of Logarithms, 1889. 3 For a description of Napier's bones, see article "Napier, John," in the Encyclopedia Britannica, 9th Ed. In the dedication Napier says, " I have always endeavoured according to my strength and the measure of my ability to do away with the difficulty and tediousness of calcula tions, the irksomeness of which is wont to deter very many from the study of mathematics." See MACDONALD'S Ed. of NAPIEE'S Construction^ p. 88.
MODERN TIMES
157
takes pains to explain multiplication, division, and evolution by "Neper's Bones or Bods." His logarithms were the result of prolonged, unassisted metic,
and isolated speculation. x x is the logarithm n b ,
Nowadays we usually say n
of
to the base
But
6.
that, in
in the time
exponential notation was not yet in vogue. The attempts to introduce exponents, made by Stifel and Stevin, were not yet successful, and Harriot, whose algebra
of Napier our
appeared long after Napier's death, knew nothing of indices. It is one of the greatest curiosities of the history of science
Napier constructed logarithms before exponents were That logarithms flow naturally from the exponential
that used.
1 symbol was not observed until much later by Euler. What, line was of ? then, thought Napier's Let AE be a definite line, AD' a line extending from A Imagine two points starting at the same moment; indefinitely. 1
A
the one moving from toward E, the other from
A
along
AD
f .
i
,
j^
^
^
,
,
Let the "
velocity during the first
moment be the same
'
,
for
AD
be uniform; but Let that of the point on line the velocity of the point on decreasing in such a way both.
1
AE
when
that
arrives at
any point
AC
it
AC
1
,
1
us develop the theory more fully. velocity = AE = (say) interval or
v.
Assume a very
large initial
Then, during every successive short
moment of time, measured approximately by
~,
the
v
1 J. J. WALKER, " Influence of Applied on the Progress of Pure Mathematics," Proceedings Lond. Math. Soc. t XXII., 1890.
A HISTORY OF MATHEMATICS
158
lower point will travel approximately unit distance,
wMch
is
the product of the uniform velocity v and the time  (nearly),
The upper
point, starting likewise with
travels during the first
AB, and
arrives at
moment very
a velocity v
= AE,
nearly unit's distance
B with a velocity = BE = v
1
 \
=?/ 1
During the second moment of time the velocity of the upper point
is
very nearly v
1,
hence the distance
BO is ^ "~.
.,
v
and
= wl. The = BJSr JBO=w lv \ vj distance of the point from E at the end of the third moment and after the if* moment, is similarly found to be t/ 1 the distance (ZB
j
w 1 
.
'The distances from
1
end of successive moments first
of the two following
0,
1,
2,
,
E
of the upper point at the
are, therefore, represented
by the
series (approximately),
3,
,
v.
The second series represents at the end of corresponding from A** intervals of time very nearly the distances of in lower series are ap the the numbers to According Napier,
D
1
proximately the logarithms of the corresponding numbers in the upper series. Kow observe that the lower series is an arithmetical progression and the upper a geometrical progres It is here that Hapie^s discovery comes in touch with sion. tlte
work of previous
it is
investigators, like
Archimedes and
Stifel
here that the continuity between the old and the
The
relation
between numbers and their logarithms,
;
new
wMdb
159
MODEBH TBtES
indicated by the above series, is found, of course, in the loga rithms now in general use. The numbers in the geometric is
series, 1, 10, 100, 1000,
have for their common logaxithms
(to
the base 10), the numbers in the arithmetic series, 0, 1, 2, 3. Bat observe one very remarkable peculiarity of Napier's log arithms: they increase as the numbers themselves decrease and numbers exceeding t? have negative logarithms. More zero is the logarithm, not of unity (as in modern log over,
to
which was taken
10 7
by Napier equal arithms), but of v, of successive integral Napier calculated the logarithms, not but of sines. His aim was to sim from 1 upwards,
numbers,
.
AE
was the sine The line plify trigonometric computations. 7 of 90 (i.e. of the radius) and was taken equal to 10 units. and AB', A'C', AD' their from what has been respective logarithms. said that the logarithms of Napier are not the same as the natural logarithms to the base e = 2.718. This dif
BE, CE, DE, were
sines of arcs,
It is evident
must be emphasized, because
ference
is
it
not
uncommon
for textbooks OB algebra to state that the natural logarithms 1 The relation existing between were invented by Napier, natural logarMims and thoe of Napier is expressed by the
formula/
It mast; be mentioned that; Napier *
IB view of tke fact that
tmy were
Hie
first
Genoaa
to point out
KomerM&ms
tiais
iM
writers of
not dBtermiBe the
tJ*e
diSermee,
close of tlie last een
it is
curloiis t& find
"
m
I^garitikmBS," tfoe For references to na&iral in?eated thai statement logarithms. Kapler articles by eaaiy writers pointing out this error, coasult I>R. S. YertAisc&te Untersw&ungen, Chap. V., or my Teaching
Broekk&n^
L&eUco* (1894),
article
md
Mathematics im 2
For
its
&e
United Sl^as, p. 390.
derivation see CAJOKI, History of Ma&em&&cs, p. 168.
A
160
HISTORY OF MATHEMATICS
Ms system of logarithms. The notion of a "base" not only never suggested itself to him, but in feet is inapplica 1 ble to Ms system, unless it is modified somewhat.
base to
Napier's great invention was given to the world in 1614, in a work entitled, Mirifici logaritkmorum canonis descrip&o* In it
he explained the nature of logarithms, and gave a logarith
mic table of the natural sines of a quadrant from minute to In 1619 appeared Napier's Mirifici logarithmorum minute. canonis
constructio,
as a
posthumous work,
method of calculating logarithms
is
3
explained.
wMch
in 4
The
his
follow
"
"
That the notion of a base may become applicable, it is necessary This relation log 1 = that zero be the logarithm of 1 and not of 107 B obtained approximately if each term of Napier's geometric series and 1
.
each term of his arithmetic series
>
('AM )
o
Here
1
to 61
,
is
divided by v or 1C7 .
('
JL
JL
_L
10*'
10''
10"'
appears as the logarithm of
where
e
= 2.718
.
This gives us
f
1

,
which
is
nearly eqnal
Hence the base of Napier's logarithms, as
here modified, is the reciprocal of the base in the natural system. u Since the caka1 From a note at the end of the table of logarithms : table, which ought to have been accomplished by the labour and assistance of many computers, has been completed by the strength and I ndostry of one alone, it will not be surprising if many errors have crept into it" The table is remarkably accurate, as fewer errors have been found t&sw might be expected. See NAPIER'S Construction BONALB'S IkL), $>f>. 87, 906.
iatkm of this
* It
tkm 4
has been r^mMiBbed in Latin at Paris, 1895. An English by W. E, Macdonald, appeared in
1
of the CoRStrvctio,
For a
brief explanation of Napier's
mode
of compatation see
MODERN TIMBS ing is a copy of part of 1614;
At
the bottom of
right, is
u
161
page of the Descriptio of
tlie first
the first page in the DescriptiOy on the In the columns marked
the figure "89," for 89.
we have here copied the natural sines of 0, to 5 of 89, 55 to 60 minutes. In the columns marked and minutes, u logarithmi," are the logarithms of these sines, and in tiee column u diffieremtisB ** the differences between the logarithmic sinus,"
figure
m fee
twi>
oteans.
Since sina; = cos (90
a?),
this
semiquadrantal arraagement of the tables really gives all the cosines of angles and their logarithms. Thus, logcosO5
f
= 11 aad^oe^
p 5i f
= 6^SlM5*
M^reorer, slaee log tanx
gives the logariliimie
cotangents if
tafcemt
if
4, and the
.
Ic^arithms mefo witii immediate IB England and on the continent. W!M> in Napier's time was professor of geomefey IB
Hem
Lcdon, and afterwards
wm
professor at Oxford, w Neper, lord of Mark
witti adBairafci
lite IMct. o/ National Bio^rapKy gives 1561 as the dsute of his birth.
A HISTOBT OF MATHEMATICS
162
inston, hath, set
my
head and hands at work with his new and
admirable logarithms. I hope to see him this summer, if it please God, for I never saw a book which pleased me better
and made me more wonder.''" Briggs was an able mathe matician, and was one of the few men of that time who did not believe in astrology. While Kapier was a great lover of this pseudoscience, " Briggs was the most satirical man against it
that hath been known," calling
conceits."
Briggs
it
"a
left his studies in
to the Scottish philosopher.
system of groundless to do homage
London
The scene
at their meeting is
and Napier Briggs was interesting. complained to a common friend, "Ah, John, Mr. Briggs will not ooine." At that very moment knocks were heard at the delayed in his journey,
and Briggs was brought into the lord's chamber. Almost onequarter of an hour went by, each beholding the other gate,
without speaking a word. At last Briggs began " My lord, I have undertaken this long journey purposely to see your :
person,
came
and to know by what engine of wit or ingenuity you to think of this most excellent help in astronomy,
first
my lord, being by you found out, I out before, when now known it is so easy." l Briggs suggested to Napier the advantage that would result from retaining zero for the logarithm of the viz.
the logarithms ; but,
wonder nobody found
whole
sine,
part of that
it
but choosing 10 7 for the logarithm of the tenth sine, i.e. of 5 44 22". Kapier said that he
same
f
had already thought of the change, and he pointed out a slight improvement on Briggs's idea viz. that zero should be the logarithm of 1, and 1C7 that of the whole sine, thereby ;
characteristic of numbers greater than unity and not negative, as suggested by Briggs. Briggs positive admitted this to be more convenient The invention of "Brig
naaking the
i
MAmtH^FfBii's Memoirs of Jtiw Jfo^er, 1834,
p.
4m.
MO0BBH TIMES gian logarithms
"
occurred, therefore, to Briggs
The
independently.
system was that dated to the base,
163
its
fundamental progression was accommo our numerical scale. Briggs devoted
10, of
his energies to the construction of tables
ail
and Kapler
great practical advantage of the new
upon the new
Kapier died in 1617, with the satisfaction of having found in Briggs an able friend to bring to completion his plan.
In 1624 Briggs published his Arithmetic logarithmica, containing the logarithms to 14 places of num The gap bers, from 1 to 20,000 and from 90,000 to 100,000. unfinished plans.
from 20,000 to 90,000 was filled by that illustrious suc cessor of Napier and Briggs, Adrian Vfaeq, of Gouda in Holland. 1
to
The
He
published in 1628 a table of logarithms from
100,000, of
which 70,000 were calculated by himself.
publication of Briggian logarithms of trigonometric functions was made in 1620 by .Edmuud Guiiter, a colleague of Briggs, who found the logarithmic sines and tangents for first
every minute to seven places. Gunter was the inventor of the words cosine and co&mgeni Briggs devoted the last years of Ms life t eaieokfeig more extensive Briggian logarithms .
of
trigoiM3iii0tarie
fm^Mm^
but
lie
died in 1630, leaving
Ms
was oamad on by the English Hetwy Ge&ibrcuML, and then published by Ylacq at Ms own expense. Briggs divided a degree into 100 parts, font owing to the
work
unfeiis&ed.
It
publication by Ylaeq of trigmK^&oeal tables constructed 0n the old sexagesimal division^ the innovaiion of Briggs remained unrecognized. Briggs and Viaeq published fetor
fundamental works, the results of
wMek "have
never been
"1 superseded by any subsequent calculations. 1
For further information regarding logarithmic
tables, consult the
in the
in the English Cyclop&dia, in the Penny C^lopvdia, and J. W. Iu GLAISHKB in the report of the committee on mjih^my44^ tables,
A HISTORY OF MATHEMATICS
164
We
have pointed out that the logarithms published by
Napier are not the same as our natural logarithms. The first appearance of natural logarithms (with the decimal point omitted) is in an anonymous Appendix printed in the 1618 edition of Edward Wright's translation of Napier's Descriptio. This interesting Appendix explains methods of interpolation
and
very probably written by William Oughtred.
is
These
interpolations were effected with the aid of a small table con
taining the logarithms of 72 sines.
230 2584.
In modern notation,
loge
In
it,
log 10
is
given as
10 = 2.302584.
It is frequently stated that natural logarithms appear in the New Logartthmes, published at London by a London teacher of
mathematics, John Speiddl, in the year 1619. tains logarithms of sines, tangents, and secants.
The book con These are not
natural logarithms, on account of complications aris ing from the fact that the logarithms appear in the tables as
quite
tiie
numbers and
integral
also that the natural functions (not
by Speidell) were taken as integral numbers. 7 Napier gives sin 30 = 87265, the radius being 107. In printed
mn
= .0087265.
3CK
The
Thus, reality,
natural logarithm of this fraction
is
approximately 5.25861. Adding 10 gives 5.25861. Speidell writes in his tables, log sin 307 525 861. The relation be
=
tween the natural logarithms and the logarithms in SpeidelFs trigonometric tables is shown by the formula, Sp. log a?
=
f loge
10^10
For secants and the later Mlf of the tan
jgA
gents the addition of 10 is omitted. In SpeideiPs table log 89 404 812, the natural logarithm of tern 89 being
=
fern
4.04812. 1
puMisbed In the Report of the Brtti&k AssKtc&xtion for ike Advancement
^memcefor 187&, *
4$,
On
m&
i^^^m
p>.
SiisiBfci^ see
1175.
^rteffy Jvrnnea o/Pwrs md Appl.
Mies 'm the EnpMsk pp. 174178; aitlele Tafelee in tke Mej*ft &tl*e British Axsctot f
9
t
JfiatfL,
V*
CycLop^Sm; Be
MODERK TBdS From
J
Speidell s statements
166
evident
it is
bat purely practical reasons induced
tiiat
Mm
to
no
theoretical,
modify Napier's
He
desired to simplify logarithms, so that persons ignorant of the algebraic rules of addition and subtraction
tables.
His tables of 1619 aYoid negative num important adaptation to the Arabic Notation, brought about in the Briggkn system, escaped him. His son, Euclid Speidell, says that he " at last concluded that the deci could use the tables.
bers, but
titie
mal or Briggs' logarithms were the best sort for a standard logarithm." Editions of SpeidelPs book appear to have been issued in 1620, 1621, 1623, 1624, 1627, 1628, but not all by himself. In Ms " Briefe Treatise of Sphsericall Triangles " he mentions and complains of those who had printed his work without an atom of alteration and yet dispraised it in their
To them he
prefaces for want of alterations.
" If that thou canst amend it So shall the arte increase If thou canst not commend
says
:
:
:
it,
Etoe, preetbee hcmid thy peace."
Thm HuMr
fereatacnt of himself Speidell attributes to his
aot leafing been at Oxfocd or Cambridge
In the edition of
*
" not
Ms New Logmtkmes,
hatting
pub
Speidell
mmimm 11W0; Ubese aare nafeirai inely logaritii^as, mm$& fm tfe O^OB of tito mal point We have mm&&&e& fee ^mj mmM Mile of lished also logarife
f
Ute
which contains 72 logarithms.
SpewMPs
is tiie earliest
more
extensive printed table of natural numbers. There is a eolmmi in the laMe which shows that he means to use his table cal
m
culations
by
feet,
inehe, and quarters.
Thus tto number 77B
has 1614 opposite to it, there being 7T5 quarter inciies in 16 feet 1 inehj S quartos. This is an interesting example 0f the eiforts B^tde feom time to iime to partially OTe^oomfee fee
A HISTORY OF MATHEMATICS
166
inconvenience arising from the use of different scales in the system of measures from that of our notation of numbers. At the bottom of each page Speidell inserts the logarithms of 100 and 1000 for use in decimal fractions. The most elaborate system of natural logarithmic tables is Wolfram's, which practically gives natural logarithms of all numbers from 1 to 10,000 to 48 decimal places. They were 1 Wolfraip published in 1778 in J. C. Schulze's Sammlung, was a Dutch lieutenant of artillery and his table represents six years of very toilsome work. The most complete table of
natural logarithms, as regards range, was published by the (xerman lightning calculator Zacharias Base, at Vienna, in
A
found also in Rees's Cyclopaedia (1819), Hyperbolic Logarithms." In view of the fact that all early efforts were bent, not
1850.
article,
table
is
"
toward devising logarithms beautiful and simple in theory, but logarithms as useful as possible in computation, it is curious to see that the earliest systems are comparatively weak in practical adaptation, although of great theoretical interest.
upon natural logarithms, whose modulus while Speidell luckily found the nugget. only possible rival of John Napier in the invention of
ITapier almost hit is unity,
The
logarithms was the Swiss Joost
Burgi or Justus Byrgius a In his watchmaker, he afterwards was youth (16521632). at the observatory in Kassel, and at Prague with Kepler. He
was a mathematician of power, but could not bring himself to publish his researches. Kepler attributes to him the discovery of decimal fractions and of logarithms. Burgi published a crude table of logarithms six years after the appearance of Paper's J9esenpi0, but it seems that he conceived the idea aail constructed that table as early, if not earlier, than Napier
English
HOBEEH However, he neglected to have the results published until after Napier's logarithms (largely through the influence of Kepler) were known and admired throughout Europe. did
his.
1
The methods
of computing logarithms adopted by Napier, from the Brigga, Kepler, Ylaeq, are arithmetical and deducible doctrke of ratios. After the large logarithmic tables were
once computed, writers like Gregory St. Vincent, Newton* Nicolaus Mercator, discovered that these computations can be In performed much more easily with aid of infinite series. the study of quadratures, Gregory St. Vincent (15841667) in 1647 found the grand property of the equilateral hyperbola
which connected the hyperbolic space between the asymptotes with the natural logarithms, and led to these logarithms being called
MereatoT in hyperbolic/' By this property ; Nicolaus at the logarithmic series, and showed how the
i
1668 arrived
construction of logarithmic tables could be reduced to the quadrature of hyperbolic spaces.
by
series
2
English Weights <md Measures
At as k&e a period as eomiaeree
the sixteenth century, the condition of Owing to men's ignorance
m Epglswi was very low.
sad tfaa general barbarism of the times, interest on in the thirteenth entay mounted to an enormous rate.
of trade
money
1
But in course of the next two eateries a reaction followed. Not only were extortions of this sort prohibited by statute, but, in tfete reign of Heary VTL y serere laws were made against taking any interest whafcerer,
Instances occur of 50
% rates.
1 For a description of it see GhmHiJiiwr's &ex$i. 1877, p. 119 ; see also p, 75. " 2 For Tanoas methods of computing logarithms, eoiasiilt art
nihiBs" in
tJbfi
&&c&d0&8$M BH&tnnfaa,
9tfe
HUME'S History of Mnj^m^ Ckap. XIL
Ed.
A
168 all interest
HISTOBY OF MATHEMATICS
being then denominated usury.
did not nourish under these
new
conditions
That commerce is
not strange,
middle of the sixteenth century we find that the invidious word usury came to be confined to the taking of
But
after the
exorbitant or illegal interest; 10% was permitted. What trade existed up to this time was carried on mainly by
little
merchants of the Hansa Towns called.
But in the
latter half of the fifteenth century the use of
gunpowder was
introduced,
America discovered. to quicken.
Even
commerce became
the
art
of
printing invented,
The
pulse and pace of the world began in England the wheels of commerce were
gradually set in motion. for
Easterlings as they were
1
business careers.
In the sixteenth century English felt of some preparation
Need was
brisk.
Arithmetic
and bookkeeping were
introduced into Great Britain,
Questions pertaining to money, weights and measures must of necessity have received some attention even in a semiciv ilized
community.
In attempting to sketch their history, we and measures, but also in English Roman influence. The Saxons improved
find not only in weights coins, traces
of
Boman
coinage. That of William the Conqueror was appar on the plan adopted by Charlemagne in the eighth ently and is supposed to be derived from the Romans as century
regards the division of the pound into 20 shillings and tlw The same proportions were preserved shilling into 12 pence. in the lira of Italy, libra of Spain, and Wwre of France, all
now
obsolete. The pound adopted by William the Gcmqueror was the Saxon Mbneyefs or Tower pound of 5400 grains.1
Hres
is
England, Cba$>, XXXV. Kmixr's Uxiwsrml Oam&tei^ Londo^ 1835, Vol. L, f our information regarding Kftiytlah money, weights, and i derived from this work.
MODEKH TiMre
169
Notice that the vegetable kingdom supplied the primary unit of u mass, the grain." A pound in weight and a pound in tale (that
An amount of silver weighing one pound was taken to be worth one pound in money. This IB reckoning) were the same.
is,
" pound," first as a explains the double meaning of the word Later the Troy poaad weightunit, secondly as a moneyunit
(5760 grains) came to be used for weighing precious meials, Troy pound of silver would therefore contain 2I of fee
A
mentioned above. Between the thirteenth century and the beginning of the sixteenth the shilling was reduced to  its ancient weight A change of this sort would probably have been disastrous, except for the practice which appears to shillings
have been observed, of paying by weight and not by tale. If we except one short period, we may say that the new shilling
was permitted to vary but little in its fineness. In 1665, in the reign of Charles II., a Troy pound of silver yielded 62 shillings, in
1816
it
1 yielded 66 shillings.
A
financial
expe
dient sometimes resorted to by governments was the deprecia tion of the eorreney, by means of issues of coin which contained
muck
o%
lass silver
r gold than older coins, but
TMs
which had nomi
by Henry Yin. ? who issued moB^y in wLkh the silver was reduced in amount by , later , and finally f* IB tbe reign of Edward YL, the amount of silver was reduced by f? so that & coin contained only the old amount of silver, Seventeen years after t&e first of these issues Queen Elizabeth called in the base coin aad pot money of Oie old character in cbradatioa. Henry VIII/s experiment was disastrous, and reduced Tfriglarai "from the positkxa of a first rate to that of a third rale power in Europe for more tfeaa Ibe same value.
was
tried
aeentey.*** *
*
also
Kmuwr, YoL i,
p.
m
"Hnaaoe,* m Mw^cMop^ia ^rU&^^ Q^m,, FRJJ*CIS A. WALKER, Moaey, Chaps. X. aad XL
Article,
Consult
A HISTORY OF MATHEMATICS
170
Interesting are the etymologies of some of the words used in connection with English money. The name sterling seems to have been introduced through London. " In the time of ...
the
Hansa merchants
King Biehard
I.,
in
monie coined
in the east parts of Germanic began to be of especiall request England for the puritie thereof, and was called Easterling monie, as all the inhabitants of those parts were called Easkerin
and shortly after some of that countrie, skillful in mint were sent for into this realme to bring the coine matters^ to perfection which since that time was called of them sterling, UngSj
.
.
.
;
for Easterling" (Camden). In the early coinages the silver penny or sterling was minted with a deep cross. When it was
broken into four parts, each was called & fourthing or farthing, the ing being a diminutive. Larger silver pieces of four pen&e were first coined in the reign of Edward III. They were called greats or groats.
In 1663, in the reign of Charles EL,
new gold
coins were issued, 44 pieces to one Troy pound of the metal. These were called guineas, after the new country
on the west coast of Africa whence the gold was brought. 1 The guinea varied in its current price from 20 shillings up to 30, until the year 1717, when,
mendation,
The
it
was fixed
on Sir Isaac Newton's recom
at 21 shillings, its present value, 2
history of measures of weight brings out the curious
among the Hindus and Egyptians, as well as Italians, other Europeans, the basis for the unit of weight and English, in the grain of barley. This was also a favourite lay usually unit of length. The lowest subdivision of the pound, or of fact that
1
THOMAS DILWOETH
abottfc
Mony
in his Schoolmaster's Assistant, 1784 (first ed&io
Sss of
1743) praises English gold as follows, p. 89 : "In England, axe paid in the best Specie, yiz. , Guineas, by which Means 1$
1
pat into a small Bag, and conveyed away in tae Pfcdoefc; b in Sweden they often pay Sams of Mony in Copper, and fee ofeant is obliged to send Wheelbarrows instead of Bags to receive Ik 1 or more
may be
MODERN TIMES
171
other similar units, was usually denned as weighing the same That no great degree as a certain number of grains of barley. of accuracy could be secured and maintained on such a basis The fact that the writings of Greek physicians is evident
were widely studied, led to the general adoption in Europe of the Greek subdivisions of the lUra or pound. The pound contained 12 ounces; i4ie lower subdivisions being the drachm or draw ("a handful"), the gramma ("a small weight"), and The Bomaas taranslated gramma, into scripiidum the grain. 1 or scrufftdum, which word has eme down to us as " scruple."
The word gramme has been adopted into tfce metric system. The Greeks had a second pound of 16 oanee^ ealled mina. The subdivision of pounds both duodecimally and according power of two
is
therefore of aneiait date.
3>uring the Middle Ages there
was in Europe an almost
to the fourth
endless variety of different sizes of the pound, as afeo of the " " " " foot, but the words pound and foot were adopted foj aH languages ; thus pointing to a common origin of the measures.
The
various pounds were usually divided into 16 sometimes into 12. Th word a pound n conies from the
The word * mwe n
jxmdss. twelftia
part"
Tfae wotis
is
the Latin smeia, meaning u a
^onnee 77 and
"m&k"
have one
coninion derivation^ the former being ealled undo* Wbrm (libra, poiuad),
and the
latter undo, pe$i& {p&fy foot).*
Chi English standards of weights
and
mmm&m HiBre exited
(we are told by an old bishop) go4 laws befot tot the laws then, as subsequently^ were mot well
%
The Saxon Tower pound was retained WiEiam queror and ^rv^i at first both as monetary and as weight ?
unit.
There have been repeated alterations in
pounds in use in England. 1
PBJLCOCK, p. 444.
tlie size
Moreover, several different *
KJBIXT,
YoL
L, p. 20.
of tto
Muds
A HISTORY OF MATHEMATICS
172 were
The
in use for different purposes at
one and the same tima
serious attention to this subject seems to have been " in 1266, statute 51 Henry IIL, when, by the consent of given first
the whole realm of England
... an English penny,
called a
round and without any clipping, shall weigh 32 wheai corns in the midst of the ear, and 20 pence do make an ounce, and 12 ounces one pound, and 8 pounds do make a gallon of
sterling,
wine, and 8 gallons of wine do 77 is the eighth part of a quarter.
make a London
bushel, which
According to this statement,
the pound consisted of, and was determined by, the weight of 768Q grains of wheat Moreover, silver coin was apparently taken by weight, and not by tale. Ho doubt this was a grievance, when the standard of weight was so illestablished. Ute pound defined above we take to be the same as the Tower poond. In 1527, statute 18 Henry VUL, the Tower pound,
wMeli had been used mostly for the precious metals, was The earliest afooli&bed, and the Troy established in its place.
Troy pound is one of 1414, 2 Henry V. h0w much earlier it was used is a debated question. The old Tower pound was equivalent to ^f of the Troy* The name 2Vof is generally supposed to be derived from Troyes in France, where a celebrated fair was formerly held, and the statute mentioning the
;
pound was used. The English Committee (of 1758) on Weights and Measures were of the opinion that Troy canie from the monkish name given to London of TTG^O^^ 1 founded on the legend of Brute. AoeofdiBg to this, Troy According to mythological history, Brote was a cbsoeadaat of JSneas and having inadYertently killed Ms father, 3ed to Britain, founded Loadoo, and called it T&fnow^ewTivj). Speeserwi&es, *
of ancient Troy,
"Ifceiy
Queea"
HI., 9,
For noble Britons sproag from Trojans bold TroyiKyva^t was IwiBt of oM Troyes a^ies col^
AM
MODEEK
173
TOfflB
" weight means London weight" At about the same period, per haps, the avoirdupois weight was established for heavy goods.
The name
is
commonly supposed
to be derived
from the French
&0wr~dMrpQi&i a corrupt spelling introduced for %mrdepoi&, and signifying "goods of weight" 2 In the earliest sfcatetes
in which the word avoirdupois is used (9 Edward IIL, in 1335 j and 27 Edward IIL, in 1353), it is applied to the goods tibemThe latter statute say% selves, not to a system of weights. " Forasmuch as we have heard that some merchants purchase
avoirdupois woollens and other merchandise by one weight and sell by another, . . we therefore will and establish that
one weight, one measure, and one yard be throughout the land, and that woollens and all manner of avoirdupois be
.
.
.
weighed. "
" .
.
.
In 1532, 24 Henry VIII., it was decreed that and veal shall be sold by weight called Here the word designates weight According arithmetic of entitled " The of
beef, pork, mutton, s
haverdupois." to an anonymous
Pathway
1596,
u
Knowledge," the pound haberdepois is parted into 16 ounces ; every ounce 8 cbagmes, every dragme 3 scruples, every scruple
20 gaina."
IFMs pcraad eoatams the same as the statute pOTiM of 1266 and the grains
number (7680) of same subdivisions
of the oonoe, drachm, aad scruple as our present apothecaries' weight The number f grains im fee pennyweight of the old
pound was changed from 32 to 24, reBderiag the number 01 graiBS in the pound 5760, Why ant ivbeR iMs cfeaage was made we do not know. Codber, Wii^afe, ^a, say in Hbek arithmetics tiiat
Tim cmgia
*
32 gmim* of wheat
of our apoiheeairies*
COCXB*S ArOMmetfe,
make 24 artijteal grain*."* has been
ira^^ it
DmJ^lm, 1714, p. IS
1^ Id., 17S&, p.
;
7.
and SfflKLir^ BdL
Gmm^B
A HISTORY OF MATHEMATICS
174
was this Medicines were dispensed by the old subdivisions of the pound (as given in the "Pathway of Knowledge") and continued to be so after the standard pound (with 24 instead of 32 grains in the pennyweight), which Queen Elizabeth :
ordered to be deposited in the Exchequer in 1588, supplanted There appear, thus, to have been two different
the old pound.
The new of Queea pounds avoirdupois, an old and a new. Elizabeth agreed very closely with, and may have originated 1 It must be remarked that from, an old mercitant's pound. before the fifteenth century, and even later, the commercial weight in England was the Amsterdam weight, which was
then used in other parts of Europe and in the East and West 2 Indies. In Scotland it has been in partial use as late as our eentury, while in England it held its own until 1815 in
fcdng assize of bread.*
This weight tells an eloquent story the extent of the Butch trade of early times. regarding Other kinds of pounds are mentioned as in use in Great Britain. 8
Their variety
is bewildering and confusing ; their Mstories and relative values are very uncertain. In fact a pound of the same denomination often had different values
in different places, The "Pathway of Knowledge," 1596, gives five kinds of pounds as in use: the Tower, the Troy, the " haberdepoys," the subtill, and the foyle. The subtill pound was used by assayers; the foyle was f of a Troy and was used
and for
for gold
foil,
wire, the
workman probably secured Ms
pound
wire,
at the price of a
pearls.
pound of
In ease of gold profit
bullion.
foil
by selling
Many
and of
a
varieties of
measure arose from the practice of merchants, who, instead of 1 **
2
" Weights and Measures in Penny or in English O^eZop^dfo.
VoL IL, p, 372, note. Ttas Dn,woftTH, in his Schoolmaster's Asste&mt, 1784, p. 38, says : ** Saw, Long, Sfeort, China, Morea^Silk, ate., aa?e wigfced by a great Ponod of 24 oz. Bm Ferret, IBoedia, SleeveSilk, etc., by the Pocmdof 16 re." K.BLLT,
*
**
MODERN TIMES
175
varying the price of a given amount of an article, would vary the amount of a given name (the pound, say) at a given price. Ancient measures of length were commonly derived from 1
some part of the human body.
This
seen in cubit (length of forearm), foot, digit, palm, span, and fathom. Later their lengths were denned in other ways, as by the width (or length) of barley corns, or by reference to some arbitrarily chosen is
by the government. The cubit than the foot. It was used by greater antiquity the Egyptians, Assyrians, Babylonians, and Israelites. It was standard, carefully preserved
is of
much
employed in the construction of the pyramid of Gizeh, per The foot was used by the Greeks and the haps 3500 B.C. Romans. The Eoman foot (= 11.65 English inches) was some times divided into 12 inches or uncice, but usually into 4 palms (breadth of the hand across the middle of the fingers) and each palm into 4 fingerbreadths. Foot rules found in Roman 2
ruins usually give this digital division. By the Romans, as also by the ancient Egyptians, care was taken to preserve the standards, but during the Middle
Ages there arose great diver
sity in the length of the foot.
As
previously stated, the early English, like the Hindus and
Hebrews, used the barley corn or the wheat corn, in determin ing the standards of length and of weight. If it be true that
Henry
ordered the yard to be of the length of his arm, then It has been stated that the Saxon yard
I.
an exception. was 39.6 inches, and this is
by Henry
I.
that, in the year 1101, it
to the length of his arm.
The
was shortened
earliest
English
statute pertaining to units of length is that of 1324 (17 Edward that " three barley corns, round and IL), when it was ordained
dry, make an inch, 12 inches one Here the lengths of the barley corns 1
"
2
DE
foot, three feet
a yard."
are taken, while later, in
Weights and Measures" in English Cyclopaedia. MOKGJLN, Arith. Books, p. 5.
A HISTOEY OF MATHEMATICS
176
the sixteenth century, European writers take the breadth; " 64 breadths to one " geometrical foot (Clavius). The breadth
was doubtless more definite than the length, especially as the word " round " in the law of 1324 leaves a doubt as to how much of the point of the grain should be removed before it can be so called. It would seem as though uniformity in the standards should be desired by all honest men, and yet the British government has always experienced great difficulty in enforcing it. To be
provisions of law often increased the confusion. by statute 15 Henry VI., the alnager, or measurer " by the ell, is directed to procure for his own use a cord twelve yards twelve inches long, adding a quarter of an inch to each sure, the
Thus, in 1437,
This law marks the era when the woollen became important, and the law was intended to
quarter of a yard." maeudfewjfcure
make
certain the hitherto vague custom of allowing the width
of the
thumb on this, it
every yard, for shrinkage.
was ordained that " cloths
In 1487, as
if
to
shall be wetted before
repeal they are measured, and not again stretched." 1 year the older statute is again followed.
But
in a later
There was no defined relation between the measures of length and of capacity, until 1701. The statute for that year declares that "the Winchester bushel shall be round with a
plane bottom, 18 inches wide throughout, and 8 inches deep." In measures of capacity there was greater diversity than in the units of length and weight, notwithstanding the fact tha& in the laws of King Edgar, nearly a century before the Con
gest, an injunction was issued that one measure, the Winches ter, should be observed throughout the redbn. Heavy fines were imposed later for using any except the standard without avaiL The history 2 of the wine gallon il
ifoet
1
JT0r& j&mowGm Memtm* Ho, XCYIL, October, American Memem* H. XCVU.
* North.
MODERK TIMES
177
how
standards are in danger of deterioration. By a statute of Henry III. there was but one legal gallon the wine gallon.
Yet about 1680
it
was discovered that
for a long time importers
of wine paid duties on a gallon of 272 to 282 cu. in., and sold the wine by one of capacity varying from 224 to 231 cu. in.
The British Committee of 1758 on weights and measures seemed to despair of success in securing uniformity, saying "that the repeated endeavours of the legislature, ever since Magna Charta, to compel one weight and one measure through out the realm never having proved effectual, there seems to be expected
shown The
little
from reviving means which experience has
to be inadequate."
latter half of the eighteenth and the first part of the nineteenth century saw the wide distribution in England of standards scientifically defined and accurately constructed.
Nevertheless, as late as 1871 it
was stated
in Parliament that
in certain parts of England different articles of merchandise were still sold by different kinds of weights, and that in
Shropshire there were actually different weights employed same merchandise on different marketdays. 1
for the
The
early history of our weights and measures discloses the fact that standards have been chosen, as a rule, by the people
themselves, and that governments stepped in at a later period and ordained certain of the measures already in use to be legal, to the exclusion of all others.
from the
Measures which grow directly
practical needs of the people
pations have usually ient dimensions.
engaged in certain occu they are of conven
this advantage, that
The furlong ("furrowlong")
is
about the
the gallon and hogshead have dimensions which were well adapted for practical use ; shoe
average length of a furrow;
makers found the barley corn a not unsuitable subdivision of 1
JOHNSON'S Universal Cyclop., Art. " Weights and Measures.'
x
.
A HISTOBY OF MATHEMATICS
178
A
the inch in measuring the length of a foot. very remark* able example of the convenient selection of units is given by
That the tasks of those who spin might be cal culated more readily, the sack of wool was made 13 tods of 28 pounds each, or 364 pounds. Thus, a pound a day was a
De Morgan
*
:
tod a month, and a sack a year. But where are the Sundays and holidays ? It looks as though the weary " spinster " was obliged to put in extra hours on other days, that she might secure her holiday. Again, "The Boke of Measuring of Sir de Benese (about 1539), suggests to Eicharde Lande," by l " The acre is four roods, 3>e Morgan the following passage :
each rood
is
ten dayeworkes, each dayeworke four perches.
So the acre being 40 dayeworkes of 4 perches each, and the mark 40 groats of 4 pence each, the aristocracy of money and that of land understood each other easily." In a system like the French, systematically built up according to the decimal scale,
simple relations between the units for time, for amount for earnings, do not usually exist This
of work done, and is
the only valid objection which can be urged against a sys like the metric. On the other hand, the old system needs
tem
readjustment of its units every time that some invention brings about a change in the mode of working or a saving of time,
and new units must be invented whenever a new trade springs Unless this is done, systems on the old plan are, if not up. ten thousand times, seven thousand six hundred fortyseven
times worse than the metric system. Again, the old mode of selecting units leads to endless varieties of units, and to such
= 1 link, 5^ yards = 1 rod, 16J feet = 1 acre, 1^ hogshead = 1 punch.
atrocities as 7.92 inches
= 1 pole, 43560 sq. The advantages
secured by having a uniform scale and measures, which coincides with that of the weights
lArmm&ical Books, ?,l&
feel
for
MODERN TIMES
178
who have given
the subject due names are whose writers Among early English identified with attempted reforms of this sort are Edmund dotation, are admitted
all
by
consideration.
Gunter and Henry Briggs, who, for one year, were colleagues in Gresham College, London. ISTo doubt they sometimes
met
in conference on this subject.
Briggs divided the degree
Gunter divided the chain of such length that " the work be
into 100, instead of 60, minutes;
into 100 links,
more
and chose
easie in arithmetick
it ;
for as 10 to the breadth in chains,
so the length in chains to the content in acres. "
Thus, ^ the
product of the length and breadth (each expressed in chains) of a rectangle gives the area in acres.
The crowning achievement of all attempts at reform in weights and measures is the metric system. It had its origin at a time when the French had risen in fearful unanimity, de termined to destroy
all their
old institutions,
and upon the
ruins of these plant a new order of things. Finally adopted in France in 1799, the metric system has during the present century displaced the old system in nearly every civilized
So easy and realm, except the Englishspeaking countries. superior is the system, that no serious difficulty has been en countered in
its introduction,
wherever the experiment has
The most pronounced opposition to it was shown French themselves. The ease with which the change the by has been made in other countries, in more recent time, is due, been
tried.
in great measure, to the fact that, before its adoption, the metric system had been taught in many of the schools.
Rise of the Commercial School of Arithmeticians in England
Owing to the backward condition
of England, arithmetic
was
In cultivated but little there before the sixteenth century. the fourteenth century the Hindu numerals began to appear in
A HISTOBY OF MATHEMATICS
180 Great Britain.
A
single instance of
their use in the
thir
teenth century is found in a document of 1282, where the word I trium is written 3 um. In 1325 there is a warrant from Italian merchants to
pay 40 pounds the ment contains Eoman numerals, but on the by one of the Italians, 13*9, that is, 1325. complete and inverted, resembling the old ;
body of the docu outside
is
endorsed
The 5 appears in 5 in the Bamberg of Boethius. The
Arithmetic of 1483, and the 5 of the apices 2 has a slight resemblance to the 2 in the Bamberg Arith metic. The Hindu numerals of that time were so different from those now in use, and varied so greatly in their form, that persons unacquainted with their history are apt to make mistakes in identifying the digits. singular practice of Mghi antiquity was the use of the old letter fj for 5. Curious
A
errors sometimes occur in notation.
or 301, for 31.
The new numerals
Thus,
X2
for 12 ;
XXXI,
are not found in the books
printed by Caxton, but in the Mgrrour of the World, issued by him in 1480, there is a woodcut of an arithmetician sitting before
a table on which are
The use
of
tablets with
Hindu numerals
in
Hindu numerals upon them. England in the
fifteenth cen
rather exceptional. Until the middle of the sixteenth tury century accounts were kept by most merchants in Eoman is
The new symbols did not find widespread accept the publication of English arithmetics began. As in Italy, so in England, the numerals were used fey mercantile houses much earlier than by monasteries and colleges. numerals.
ance
till
The
first
important arithmetical work of English authorship 1522 by Chtfkbert TonstaM (1474
iras published in Latin in
23b earlier name is known to us excepting that of 1559). Joirn Norfolk, who wrote, about 2 1340, an inferior treatise <m
aeomit of t&& numerals in England is taken from JAMES Iteoi** s article O *& Or%^i cmi History of the JV**em&1 1874, * W* W. B* BAI^,
m&ry
^
MODERN TIMES
181
Halliprogressions which, was printed in 1445, and reissued by Norfolk con 1841. well in his Rara Mathematica,j London,
founds arithmetical and geometrical progressions, and confines himself to the most elementary considerations. at Oxford, Cambridge, and Padua, and works of Pacioli and Eegiomontanus. the from freely 1 of his arithmetic, De arte supputandi, appeared in Eeprints England and France, and yet it seems to have been but little
Tonstall studied
drew
known
to succeeding
2 English writers.
The author
states that
some years previous he had dealings in money (argentariis) He read and, not to get cheated, had to study arithmetic. everything on the subject in every language that he knew, and spent
much
time, he says, in licking
what he found into shape,
ad ursi exemplum, as the bear does her cubs. De Morgan 2 this book is "decidedly the most
According to classical
which
ever was written on the subject in Latin, both in purity of The book is a "farewell to style and goodness of matter." the sciences
London.
"
A
on the author's appointment to the bishopric of modern critic would say that there is not enough
demonstration in this arithmetic, but Tonstall is a very Euclid by the side of most of his contemporaries. Arithmetical results
he gives the frequently needed, he arranges in tables. Thus, subtrac also a of form in table addition, square, multiplication 10 numbers. the of first cubes the and division and tables, tion,
For 3 f of
of
he has the notation f
H
Interesting
his discussion of the multiplication of fractions.
here premise that Pacioli (like
We
is
must
many a
present day) was greatly embarrassed
4
schoolboy of the by the use of the term
In this "book, the pages are not numbered. The earliest known work Hindu numerals are used for numbering the pages is one and KASTNER, Vol. I., printed in 1471 at Cologne. See UNGER, p. 16, 1
in which the
p. 94. 2
DE MORGAN,
8
Arithmetical Books, p. 13.
*
CANTOR, H., 476. PEACOCK, p. 439.
A HISTOBY OF MATHEMATICS
182 "
multiplication "in case of fractions, where the product is less That "multiply" means "increase" than the multiplicand. he proves from the Bible " Be fruitful and multiply, and re :
" " I will multiply thy seed as (Gen. i : 28) j plenish the earth the stars of the heaven" (Gen. xxii: 17). But how is this to In this way: be reconciled with the product of fractions?
the unit in the product is of greater virtue or significance; thus, if \ and \ are the sides of a square, then \ represents the area of the square itself. Later writers encountered the
same
difficulty,
explanation. clearness. this
He
but were not always satisfied with Pacioli's discusses the subject with unusual
Tonstall
takes
happens thus,
1x1 = ^.
it is this,
"If 1 you ask the reason why
that if the numerators alone are
multiplied together the integers appear to be multiplied together, and thus the numerator would be increased too
Thus, in the example given, when 2 is multiplied into the result is 6, which, if nothing more were done, would By seem to be a whole number ; however, since it is not the inte ger 2 that must be multiplied by 3, but f of the integer 1 that
much.
must be multiplied by f of it, the denominators of the parts are in like manner multiplied together so that, finally, by the division which takes place through multiplication of the de ;
nominators (for by so much as the denominator increases, by so much are the parts diminished), the increase of the numer ator is corrected by as much as it had been augmented more
than was right, and by this means
it is
reduced to
its
proper
value."
This dispute is instanced by Peacock as a curious example of the embarrassment arising when a term, restricted in mean ing, is applied to
wMeh 1
a general operation, the interpretation of
depends upon the kinds of quantities involved.
The
We are tartoaiafcmg Too&fcalPs Latin, qaoted by PBU.OOCK, p. 4$&
KOBEEK difficulty
which
Is
183
TTMraa
encountered in multiplication arises also
in the process of division of fractions, for the quotient is
The explanation of the paradox a clear insight into the nature of a fraction. That,
larger than the dividend. calls for
development, multiplication and division should have been considered primarily in connection with The same course must be adopted integers, is very natural.
in the historical
in teaching the young.
First
come the easy but
restricted
meanings of multiplication and division, applicable to whole In due time the successful teacher causes students numbers.
and broadening the meanings similar plan has to be followed in
to see the necessity of modifying
A
assigned to the terms. algebra in connection with exponents.
First there is given an easy definition applicable only to positive integral exponents. Later, new meanings must be sought for fractional and nega tive exponents, for the student sees at once the absurdity, if
we say
that in
arl,
x
is
taken onehalf times as a
factor.
Similar questions repeatedly arise in algebra. Of course, Tonstall gives the English weights and meas
ures
;
he also compares English money with the French, etc. worthy of remark that Tonstall, to prevent Tyndale^s
It is
translation of the
among
New Testament from
the masses,
is
spreading more widely
said to have once purchased and burned
the copies which mmainad unsoldBnt the "bishop's enabled the Tyndale, money following yea*, to bring out a second and more correct edition! all
A
quarter of a century after the first appearance of Tomwere issued the writings of JSo&er* Beoarde
stall's arithmetic,
(15101558).
Educated at Oxford and Cambridge, he ex and medicine. He taught arithmetic
celled in mathematics
at Oxford, but found little encouragement, notwithstanding
the fact that he was an exceptional teacher of this subject.
Migrating to London, he became physician to Edward
VL,
A HISTORY OF MATHEMATICS
184 and
later to
Queen Mary.
It does not
seem that he met with
adequate to his merits, for afterwards he was recompense confined in prison for debt, and there died. He wrote several at all
works, of which his algebra.
we
shall notice his arithmetic
It is stated that Eecorde
and (elsewhere)
was the
earliest
Eng
1 lishman who accepted and advocated the Copernican theory. His arithmetic, The Ghrounde of Artes, was published in 1540. Unlike TonstalPs, it is written in English and contains
The last symbol he uses to denote +, , 21In his equality. algebra it is modified into our familiar =. These three sign symbols are not used except toward the He says 2 "+ last, under "the rule of Falsehode." the symbols
whyche
betokeneth too muche, as this line, , plaine without a crosse The work is written in the form line, betokeneth too little." of a dialogue between master and Once the pupil says, pupil. " And I to authoritie witte doe youre my subdue, whatsoever
you
say, I take
that this
it
for true,"
whereupon the master
replies
too much, "thoughe I mighte of my Scholler some credence require, yet except I shew reason, I do it not desire." is
Notice here the rhyme, which sometimes occurs in the book, though the verse is not set off into lines by the printer. In all operations, even those with denominate numbers, he tests 3 the results by " casting out the The scholar complains that he cannot see the reason for the "ITo more process. doe you of manye things else," replies the master, arguing that one must first learn the art by concisely worded roles,
9V
before the reason can be grasped. 1
BALL'S Mathematics at Cambridge,
This
is certainly
sound
p. 18.
s
CJU?TOR, IL, p. 477480. * The mfmfficiency of this test is emphasized "Bat there may be given a thousand
(nay
by Cocker as follows*
infinite) false
products in
lkm, wMch after this manner may be proved to be trae, and tMs way of proving dotli Bot deserve any example," Artikme
MODERN TIMES The method
advice.
but the reason for
185
of " casting out the 9's " is easily learned,
it is
beyond the power of the young student
not contrary to sound pedagogy sometimes to teach merely the facts or rules, and to postpone the reasoning to It is
a later period.
Whoever
teaches the
method and the reason
ing in square root together is usually less successful than he who teaches the two apart, one immediately after the other. It
is,
and
we
believe, the usual experience,
both of individuals
of nations, that the natural order of things is facts first,
This view is no endorsement of mere which became universal in England after memoryculture, the time of Tonstall and Eecorde. reasons afterwards.
Eecorde gives the Eule of Three progressions, alligation, fellowship,
thinks
it
necessary to repeat
and
for the succeeding
"the golden rule"), false position.
He
the rules (like the rule of
all
three, fellowship, etc.) for fractions.
lent then,
(or
and
This practice was preva
He was
250 years.
particu
larly partial to the rule of false position (" rule of Falsehode ") and remarks that he was in the habit of astonishing his friends
by proposing hard questions and deducing the correct answer " suche children or by taking the chance replies of ydeotes in the as happened to be place." It is curious to find in Eecorde a treatise on reckoning by " whiche doth not counters, onely serve for them that cannot write and reade, but also for them that can doe both, but have not at some times their pen or tables readie with them." He mentions two ways of representing sums by counters, the Mer chant's
11
and the Auditor's account.
d., is
In the
first,
expressed by counters (our dots) thus
198 L, 19
:
= 100 + 80 pounds. '
=10 +5 + 3 pounds. =10 + 5 + 4 shillings.
=6 + 5 pence.
s.,
A
186
HISTOEY OF MATHEMATICS
Observe that the four horizontal lines stand respectively for pence, shillings, pounds, and scores of pounds, that counters in the intervening spaces denote half the units in the next line above, and that the detached counters to the left are equivalent to five counters to the right. 1
The abacus with and Italy in the
its
counters had ceased to be used in Spain In France it was used at
fifteenth century.
the time of Hecorde, and it did not disappear in England and Germany before the middle of the seventeenth century. The
method of abacal computation
is
found in the English ex
chequer for the last time in 1676. In the reign of Henry I. the exchequer was distinctly organized as a court of law, but the financial business of the crown was also carried on there.
The term "exchequer"
is
derived from the chequered cloth
which covered the table at which the accounts were made
up.
was summoned to answer for the full annual dues " in money or in tallies." " The liabilities and the actual payments of the sheriff were balanced by means of counters placed upon the squares of the chequered table, those on the Suppose the
sheriff
one side of the table representing the value of the
tallies,
warrants, and specie presented by the sheriff, and those on the other the amount for which he was liable," so that it was easy
whether the sheriff had met his obligations or not. In Tudor times " pen and ink dots " took the place of counters. These dots were used as late as 1676. 2 The "tally" upon which accounts were kept was a peeled wooden rod split ia to see
such a
way
as to divide certain notches previously cut in
it.
One piece of the tally was given to the payer the other piece was kept by the exchequer. The transaction could be verified ;
easily 1
by
fitting the
PEACOCK,
* Article
"
London,
p.
two halves together and noticing whether
410 ; Peacock also explains the Auditor's Account. " in PAi^ajLT's Dictionc? of Potit&sal
Exchequer
MODERN TIMES " the notches " tallied or not. late as 1783.
Such
tallies
187
remained in use as
1
Shakespeare lets the clown be embarrassed by a problem which he could not do without
In the Winter's Tale
(IV., 3),
counters. lago (in Othello, I.) expresses his contempt for Michael Cassio, " forsooth a great mathematician/' by calling him a " countercaster." 2 It thus appears that the old methods
Hindu numerals were With such dogged persistency
of computation were used long after the
common and general use. does man cling to the old
in
!
While England, during the sixteenth century, produced no mathematicians comparable with Vieta in France, Kheticus in
Italy, it is nevertheless true that
Germany, or Cataldi in
Tonstall and Recorde, through their mathematical works, re Their arithmetics are above the flect credit upon England.
average of European works. With the time of Eecorde the English began to excel in numerical skill as applied to money. " are " The questions of the English books/' says Pe Morgan, in and are more data, skilfully harder, involve more figures solved."
To
this fact,
no doubt, we must attribute the ready
appreciation of decimal fractions, and the instantaneous popu The number of arithmetical writers in larity of logarithms.
the seventeenth and eighteenth centuries is very large. Among the more prominent of the early writers after Tonstall and
Eecorde
are
3 :
William
Buckley,
mathematical
tutor
of
Edward VL and author of Arithmetica Memorativa (1550) Humfrey Baker, author of The WellSpring of the Sciences ;
(1562) Edmund Wingate, whose Arithmetic^ appeared about 1629; William Oughtred, who in 1631 published his Clavis ;
1
"Tallies have been at 6, but
terest payable every three 1735, p. 407. 2
PEACOCK,
p. 408.
now
months." s
at 5 per cent, per annum, the in SHELLEY'S WING ATE' s Arithmetic,
PEACOCK, pp. 437, 441,
442, 452.
A HISTORY OF MATHEMATICS
188
MathematicoB, a systematic textbook on arithmetic and algebra; Koah Bridges, author of Vulgar ArUhmeticke (1653) ; Andrew
Tacquet, a Jesuit mathematician of Antwerp, author of several books, in particular of Arithmticce Theoria et Praxis (Antwerp, 1656, later reprinted in London). Mention should be made
The Pathway of Knowledge, an anonymous work, written Dutch and translated into English in 1596. John Mellis, in 1588, issued the first English work on bookkeeping by also of
in
double entry, 1 We have seen that the invention of printing revealed in Europe the existence of two schools of arithmeticians, the algoristic school,
teaching rules of computation and commercial
and the
abacistic school, which gave no rules of but studied the properties of numbers and ratios. BoetMus was their great master, while the former school
arithmetic,
calculation,
followed in the footsteps of the Arabs. The algoristic school flourished in Italy (Pacioli, Tartaglia, etc.) and found adher ents throughout England and the continent. But it is a
remarkable fact that the abacistic school, with its pedantry, though still existing on the continent, received hardly any attention in England. ratios with its
The
laborious treatment of arithmetical
burdensome phraseology was of no practical
use to the English merchant. The English mind instinctively rebelled against calling the ratio 3 2 1 by the name of :
=
proportio mperparticularis se&quiaUera. On the other hand it is a source of regret that the successors
of Tonstall and Recorde did not observe the high standard of decided decline is authorship set by these two pioneers.
A
marked by Buckley's Arithmetica Memoratwa, a Latin
treatise
expressing the rules of arithmetic in verse, which, we take it, was mfeended to be committed to memory. are glad to be
We
i
D MOM** Arm. Books, p. ,
27.
189
MODEBN" TIMES
work never became widely popular. The was common, before the but the practice of using them was not printing,
able to say that this
practice of expressing rules in verse
invention of
common, else the number of printed arithmetics on this plan 1 would have been much larger than it actually was. Many old arithmetics give occasionally a rhyming role, but few con fine
themselves to verse.
Pew
authors are guilty of the folly
displayed by Buckley or by Solomon Lowe, who set forth the rules of arithmetic in English hexameter, and in alpha betical order.
In this connection
it
may
be stated that an early specimen
found in the Pathway of Know the present generation as the to has come down ledge, 1596, most classical verse of its kind: of the
muse
of arithmetic, first
"Thirtie dales hath September, ApriU, June, and November, Februarie eight and twentie alone, all the rest thirtie and one,'*
As a
2 close competitor for popularity is the following stanza
quoted by Mr. Davies (Key to Button's "Course") from a manuscript of the date 1570 or near it: "
Multiplication is mie vexation Division is quite as "bad,
And
The Golden Rule
And Practice
is
drives
mie stumbling
stole
me mad,"
In the sixteenth century instances occur of arithmetics writ ten in the form of questions and answers. During the seven teenth century this practice became quite prevalent both in are inclined to agree with England and Germany.
We
on the whole, an improvement on the of older practice simply directing the student to do so and so.
Wildermuth that
1
2
it is,
DE MORGAN, DE MORGAN,
Arith. Books, p. 16. Artih. Books.
190
A HISTORY OF MATHEMATICS
A question draws
the pupil's attention and prepares his mind new information. Unfortunately, the
for the reception of the
question always relates to
done as indicated.
how a thing
is
done, never
why
it is
deplorable to see in the seventeenth century, both in England and Germany, that arithmetic is reduced more and more to a barren collection of rules. The It
is
sixteenth century brought forth some arithmetics, by promi nent mathematicians, in which attempts were made at demon stration. Then follows a period in which arithmetic was
studied solely for commercial purposes, and to this commercial school of arithmeticians (about the middle of the seventeenth 1 " we owe the destruction of demon century), says De Morgan, strative arithmetic in tMs country, or rather the prevention
of its growth. It never was much the habit of arithmeticians to prove their rules ; and the yery word proof, in that science, never came to mean more than a test of the correctness of a particular operation, nines, or the like.
by reversing the
As soon
process, casting out the
as attention
was
fairly averted to
arithmetic for commercial purposes alone, such rational appli cation as had been handed down from the writers of the six
teenth century began to disappear, and was finally extinct in the work of Cocker or Hawkins, as I think I have shown rea
son for supposing it should be called. 2 From this time began the finished school of teachers, whose pupils ask, when a ques
what rule it is in, and run away, when they groir from up, any numerical statement, with the declaration that tion is given,
*Aritb. Books, p. 21. 4 ** Cocker's Arithmetic "
"
was " perused and published after Cocker's death by John Hawkins. De Morgan claims that tiie work was not writ ten by Cocker at all, but by John Hawkins, and that Hawkins attached to it Cocker's name to make it sell After reading the article " Cocker * f in the Itic&onary of National Biography,
Hawkins innocent. to account lor most
we
are confident in believing
Cocker's sudden death at an early age is snfiicieiit of his works being left for posthumous publication.
MODERN TIMES
191 1
may be proved by figures as it may, to them. Anything may be unanswerably propounded, by means of Towards figures, to those who cannot think upon numbers. anything
the end of the last century we see a succession of works, arising one after the other, all complaining of the state into
which arithmetic had fallen, all professing to give rational explanation, and hardly one making a single step in advance of
its predecessors.
"It
may
very well be doubted whether the earlier arithme
ticians could have given general demonstrations of their pro It is an unquestionable fact of observation that the cesses.
application of elementary principles to their apparently most natural deduction, without drawing upon subsequent, or what ought to be subsequent, combinations, seldom takes place at
the commencement of any branch of science. It is the work of advanced thought. But the earlier arithmeticians and alge braists had another difficulty to contend with: their fear of
own halfunder stood conclusions, and the caution with which it obliged them to proceed in extending their halfformed language." Of arithmetical authors belonging to the commercial school we mention (besides Cocker) James Hodder, Thomas Dilworth, and Daniel Penning, because we shall find later that their books were used in the American colonies. James Hodder 2 was in 1661 writingmaster in Lothbury, He was first known as the author of Hodder's London. Arithmetic^, a popular manual upon which Cocker based his their
better
known work.
Cocker's chief improvement is the use
1 The period of teaching arithmetic wholly by rules began in Germany about the middle of the sixteenth century nearly one hundred years But the Germans returned to demonstrative earlier than in England. arithmetic in the eighteenth century, at an earlier period than did the
British. 2
See UNGER, pp.
iv,
117, 137.
Dictionary of National Biography.
A HISTORY OF MATHEMATICS
192
new mode
of the called
it),
of division,
in place of
"by giving" (as the Italians the "scratch" or "galley" method
taught by Hodder. The first edition of Hoddens appeared in 1661, the twentieth edition in 1739. He wrote also The Penman's Recreation (the specimens of which are engraved
by Cocker, with whom, it appears, Hodder was friendly), and Decimal Arithmetic^, 1668. 1 Cocker's Arithmetic^ went through at least 112 editions, 2 Scotch and Irish editions. Like including Francois Barrme in France and Adam Eiese in Germany, Cocker in England enjoyed for nearly a century a proverbial celebrity, these names being synonymous with the science of numbers. A man who influenced mathematical teaching to such an extent deserves at least a brief notice. Edward Cocker (16311675) was a prac titioner in the arts of writing, arithmetic, and engraving. In
1657 he lived "on the south side of St. Paul's Churchyard" where he taught writing and arithmetic " in an
extraordinary
manner."
In 1664 he advertised that he would open a public
school for writing and arithmetic and take in boarders near St. Paul's.
Later he settled at Northampton.8
Aside from
1
Dictionary of National Biography. s " Corrected " editions of Cocker's Arithmetic were brought out in " 1726, 1731, 1736, 1738, 1745, 1758, 1767 by George Fisher," a pseudonym for MRS. SLACK. Under the same pseudonym she published in London, 1763,
The Instructor: or Young Man's
best
Companion, containing speUwhich (the twentyeighth, 1798) under In Philadelphia the etc., as above.
ing, reading, writing, and arithmetic, etc., the fourteenth edition of appeared in 1785 at Worcester, Mass., in
the title, The American Instructor : book was printed in 1748 and 1801. Mrs. Slack is the first woman whom we have found engaged in arithmetical authorship. See Btbliotheca Matkematiea> 1805, p. 75 Teach, and Hist, of Mathematics in the U. &, ;
189G, p. 12.
*PEFYS mentions him several times in 1664 w Abroad to find out ona to table with silver plates,
me
to
do
it.
So
it
in his Diary, 10th: engrave my upon my new sliding rale beingW small that Browne who made it cannot get
I got Cocker, the
famous writing master,
to
do
ifc,
and
MODEKK TIMES the absence of
and
written,
it
193
demonstration, Cocker's Arithmetic was well evidently suited the demands of the times. He all
was a voluminous
writer, being the author of
33 works, 23
calligraphic, 6 arithmetical,
and 4 miscellaneous.
metical books are:
to
Tutor
Arithmetician (1669) gives the date 1677); ;
The
arith
Arithmetick (1664); Corn/pleat Arithmetick (1678 Peacock on page 454 ;
Decimal Arithmetic^,
Artificial
Aritli
metick (being of logarithms), Algebraical Arithmetick (treating of equations) in three parts, 1684, 1685, "perused and pub lished" by John Hawkins.
In the English, as well as the French and German arith which appeared during the sixteenth, seventeenth, and
metics,
eighteenth centuries, the "rule of three" occupies a central
Baker says * in his WellSpring of the Sciences, 1562, "The rule of three is the chief est, and the most profitable,
position.
and most
excellent rule of
rules have neede of
it,
and
it
For
arithmeticke.
all
passeth
all
other
;
all
other
which the Golden
for the
it is sayde the philosophers did name it but now, in these later days, it is called by us the !Rule Rule, of Three, because it requireth three numbers in the operation."
cause,
There has been among writers considerable diversity in the notation for this rule. Peacock (p. 452) states the question " and with refer " If 2 apples cost 3 soldi, what will 13 cost ? ence to
it,
represents Tartaglia's notation thus,
Se pomi 2 I set an hour
by him

val soldi 3
to see

him design
che valera pomi 13. it all
;
and strange
it is
to see
his natural eyes to cut so small at his first designing it, and read life I could not, with it all over, without any missing, when for
him with
my
my
read one word, or letter of it. ... I find the fellow by his dis course very ingenious and among other things, a great admirer and well read in the English poets, and undertakes to judge of them all, and that
best
skill,
;
not impertinently."
some poetical ability. 1 PEACOCK, p. 452.
Cocker wrote quaint poems and distichs which show
A HISTORY OF MATHEMATICS
194
and the older English arithmeticians write
Becorde follows
as
:
Pence.
Apples.
2^_ ^^
13
3 19
answer.
In the seventeenth century the custom was as follows (Wingate, Cocker, etc.)
:
Apples.
Pence.
Apples.
2
3
13
The notation of this subject received the special and wrote who introduced the sign
Oughtred,
:
attention of
:
M, Cantor says that the dot, used here to express the ratio, later yielded to two dots, for in the eighteenth century the German writer, Christian Wolf, secured the adoption of the dot 1
In England the reason as the usual symbol of multiplication. It will for the change from dot to colon was a different one. be remembered that Oughtred did not use the decimal point England took place in the first of the eighteenth century, and we are quite sure that quarter Its general introduction in
it
was the decimal point and not Wolf's multiplication sign
which displaced Oughtred's symbol for ratio. As the Ger mans use a decimal comma 2 instead of our point, the reason 1
CANTOR,
H,
p. 721.
3
The Germans Kepler (see UNGER,
attribute the introduction of the decimal
104
comma
to
pp. 78, 109 ; GU:NTHEB, Vermisckte Untersuchungea, p. 133). He uses it in a publication of 1616. Napier, in his Rabdologia (1617) speaks of "adding a period or comma," and writes 1993,273 (see Construction, MACDONALD'S Ed., p. 89). En^ish p.
;
GERHABDT,
writers did not confine themselves to the decimal point; the comma is often used. Thus, Marian's Decimal Artihmetick (1763) and Wilder's edition
of
Newton's Universal AritkmeUck, London, 1769, use the In Kersey'* Wingate (1735) and in Dilwortn (1784),
comma exclusiTely.
MODERN TIMES for the change could not have been the
195
same in Germany
as
Dilworth x does not use the dot in multiplication,
in England.
2 but he employs the decimal point and writes proportion Before 48. once 2 4 8 16, and on another page 3 17 :
:
:
:
:
the present century, the dot was seldom used by English If Oughtred had been in writers to denote multiplication.
the habit of regarding a proportion as the equality of two ratios, then he would probably have chosen the symbol
=
The former sign was actually used for this 3 The notation 2:4 = 1:2 was brought Leibniz. purpose by into use in the United States and England during the first quarter of the nineteenth century, when Euler's Algebra and
instead of
:
:
.
French textbooks began
to be studied
by the Englishspeaking
nations.
The
rule of three reigned
metics in
Germany
supreme in commercial arith
until the close of the eighteenth century,
and in England and America until the close of the first quarter 4 An It has been much used since. of the present century. we read
1 '
but the point is the sign actually of the "point or comma, In Dodson's Wingate (1760) both the comma and point are used, Cocker (1714) and Hatton (1721) the latter, perhaps, more frequently. do not even mention the comma. 1 THOMAS DILWORTH, Schoolmaster's Assistant, 22d Ed., London,
used.
The earliest testi also, pp. 45, 123. 1784, page after table of contents monials are dated 1743. 2 In his time old and new notations were in simultaneous use, for he to keep the says, "Some masters, instead of points, use long strokes terms separate, but it is wrong to do so ; for the two points between the ;
and second terms, and also between the third and fourth terms, shew that the two first and the two last terms are in the same proportion. And whereas four points are put between the second and third terms, they serve to disjoint them, and shew that the second and third, and first and fourth terms are not in the same direct proportion to each, 8 WILDERMUTH. other as are these before mentioned." * UNGER (p. 170) says that in Germany the rule of three was preferred
first
as the universal rule for problemworking, during the sixteenth century;
A HISTOEY OF MATHEMATICS
196
important role was played in commercial circles by an allied rule called in English chairwnde or conjoined proportion, in
and in German Kett&n&atz or Reesischer Satz. It received its most perfect formal development and most extended application in Germany and the Netherlands. French rgle
conjointe,
1
Kelly
2
attributes the superiority of foreign merchants in the
a more intimate knowledge of this rule. was known in its essential feature to the Hindu
science of exchange to
The
chainrule
Brahmagupta Tartaglia ; Biese.
In
;
also to the Italians, Leonardo of Pisa, Paeioli,
to the early Germans, Johann Widmann and Adam England the rule was elaborated by John Kersey
But the one was Kaspar Franz
in his edition of 1668 of Wingate's arithmetic.
who
contributed most toward
its
diffusion
von Bees (born 1690) of Boermonde in Limburg, who migrated There he published in Ihitch an arithmetic which in French translation in 1737, and in German transla appeared
to Holland.
tion in 1739.
In Germany the Meesischer Sodz became famous.
Cocker (28th Ed., Dublin, 1714, p. 232) illustrates the rule by the example : " If 40 1. Auverd upois weight at London is equal to 36 L weight at Amsterdam, and 90 1. at Amsterdam makes 116 L at Dantzicky then how many Pounds at Dantzick are equal to 112 1. Auverdupois weight at London?" the method, called Practice, being preferred during the seventeenth cen tury, the ChainRule during the eighteenth, and Analysis (Brachsatz, or fail to observe sfm flair Schlussrechnung) during the nineteenth.
We
There the rule of three occupied a commanding position until the present century the rales of single and double position were used in the time of Recorde more than afterwards ; the ChaisRnte never became widely popular ; while some attention was always paid to changes in England.
;
Practice. 1 The writer remembers being taught the " Kettensatz " in 1873 at tibe Kantoaaschule in Chur, Switzerland, along with three other methods, tfee "Einheitsmethode" (Schlussrechnung), the u Zexlegiingsmtfeodse"
(a iotm of Italian practice)
and " Proportion."
MODEEN TIMES (p.
234)
"The Terms
Rule foregoing,
197
being disposed according to the 7th
will stand thus,
whereby I find that the terms under B multiplied together pro duce 467T12 for a Dividend and the terms under A, viz., 40 and 90 produce 3600 for a Divisor, and Division being finished, the Quotient giveth
The
129ff Pounds
at Danteicfc for the
Answer."
chainrule owes its celebrity to the fact that the correct
answer could be obtained without any exercise of the mind. Rees reduced the statement to a mere mechanism. While in the schoolroom.
was worthless for mindculture Attempts were made by some arithmetical
writers to prove
with aid of proportions or by ordinary
useful to the merchant, the rule
But
analysis.
tmsatisfaetory.
it
for pedagogical purposes those attempts proved
A rale,
more apt to lead to
ing some tkmgfet, became
errors, but requir
known in Germany as
" Basedowsche
Bege!," being rssxwBSBttal by the cteeatioiial reformer Basedow, tfeongh BO* iist given by him. Early in the present arithmetical was m?^n&mi$ed in Germany. century, teadbing
The
of $*rm Ttm ^ss^mSij driven into the and the "Sclilfi^reei^iGag^ atoined mOTe and
&&tmrtrtde
and
rule
prominence, even though Fe^loa was partial lo fee use of proportion. The "Schlnmreeioaiiiig" is
sc^ies
designate! in English by the word Analysis. f 7, what wiU 19 yards cost? One yard will
yards
3
= $ 4433; orbya modifi<^ioa
If 3 jmids ^osl out
f ^ and 19
of the abore,
"aliquot parts": 18 yards will cost f 42; one yard 4433, This u Schli^srechnung " was
and 19 yards,
by
f 2.33,
A HISTORY OP MATHEMATICS
198
for it is is, without doubt, very much older a thoroughly natural method which would suggest itself to any sound and vigorous mind, Its pedagogical value lies in the
to Tartaglia, but
;
fact that there is
no mechanism about
that
it
requires no memorizing of formulae, but makes arithmetic an exercise in It is strange indeed that in modern times such a thinking. it?
method should have been disregarded by arithmeticians dur ing three long centuries. English arithmetics embraced all the commercial subjects previously used by the Italians, such as simple and compound interest, the direct rule of
(called
three, the inverse rule of three
by Eecorde "backer
rule of three"), loss
barter; equation of payments, bills of exchange,
annuities, the rules for single
subject of tare, trett, eloff. explain the last three terms:
"
and double
We
let
Q. Which are the Allowances usually "Weight to the Buyer ? A. They are Tare, Trett, and doff.
g.
What
A. Tare
is
position,
Dilworth
made
and
m
(p.
gain,
alligation,
and the
37, 1784)
Averdupofs great
Tare? an Allowance made
to the Buyer, for the Weight of the . Box, Bag, Vessel, or whatever else contains the Goods bought. . Q. What is Trett? A. Trett is an Allowance by the Merchant to the Buyer of 4 Ib. in is
.
104
Ib.,
that
is,
Sorts of Goods. q.
What
the sixandtwentieth Part, for ,
.
Waste or
Itast, in
some
.
isCioff?
an Allowance of 2 Ib. Weight to tlie Citizens of London, on every Draught above 3 C. Weight, on some Sorts of Goods, as Galls, Madder, Sumac, Argol, etc. 1 A. Cloff
is
Q. What are these Allowances called beyond the Seas ? A. They are called the Courtesies of London; because they are not practiced in any other Place." 1 Hie term " ek>S " had also a more general meaning, denoting a smaB aflowaaee saaide on goods sold in gross, to make np for de^cieaices wiies sold in retail PBAGOCK, p. 465.
m
MOBEEN TIMBS
199
which were borrowed from the Italians, English writers added their own weights and meas ures, and those of the countries with which England traded, la the seventeenth century were added decimal fractions, which
To the above
subjects,
were taught in books more assiduously then than a century It is surprising to find that some arithmetics devoted
later.
Cocker wrote a book on have seen the rules for the
considerable attention to logarithms.
We
"Artificial Arithmetiek."
use of logarithms, together with logarithmic tables of num bers, in the arithmetics of John Hill (10th Ed. by E. Hatton,
London, 1761), Benjamin Martin (Decimal Aritkmeticfc, London, 1763; said to have been first published in 1735), Edward Hatton (Ittfire System of Arithmetic, London, 1721) and in
Edmund Wingate. who pursued mathematics for
Wingate was a London lawyer Spending a few years pastime. 1 in Paris, he published there in 1625 his Anh mMique togar r&kmique) which appeared in London in English translation in 1635.* Wingate was the first to carry Briggian logarithms About the year (taken from Gunter's tables) into France.
editions of
i
1629 he published his Arithmetic^ "in which his principal design was to otmal the difficulties which ordinarily occur in the using of logarithms
:
work into two books; the
To first
perform this he divided Ms ha ealled Naimr&l, and the
^^mdArti^eMAr^m^ie.^* Subsequent editions of Wingate two book% and were broeght out by George Shelley, and lastly by Janes by John The work was modified so largely, that WiBgaie Dodsjon. would not have known it.
rested on the first of those
Kersey, later
1
DB MOBSAS, Ar&h. Bo&ks;
Sciences Xathematiques 1628, instead of 1625.
tt
la MAXIMILIKH MAEOE'S, m*tofae dt* is giren the date
Physiques, Vol. III., p. 226,
2 MAKIE, VoL HI., p 225. JAMES DOBSON'S Pre&ee to W*NGATE'S Aritim^c, LtfBctoa, lim. The term " artificiaJ numbers" for logarit.hma is due to Napier himseit
A HISTORY OF MATHEMATICS
200
It is interesting to observe
how
writers sometimes intro
duced subjects of purely theoretical value into practical arith
Perhaps authors thought that theoretical points which they themselves had mastered and found of interest, metics.
ought to excite the curiosity of their readers.
Among
these
and higher roots, continued frac tions, circulating decimals, and tables of the powers of 2 up to the 144th. The last were " very useful for laying up grains of corn on the squares of a chessboard, ruining people by horse shoe bargains, and other approved problems" (De Morgan). The subject of circulating decimals was first elaborated by John Wallis (Algebra, Ch. 89), Leonhard Euler (Algebra, Book subjects are square, cub,
Circulating decimals were at I., Ch. 12), and John Bernoulli. one time " suffered to embarrass books on practical arithmetic,
which need have no more
to
do with them than books
<*a
mensuration with the complete quadrature of the circle/' 1 The Decimal Arithmetic of 1742 by John Marsh is almost entirely
on
this subject.
As
his predecessors
he mentions
Wallis, Jones (1706), Ward, Brown, Malcolm, Cunn, Wright After the great fire of London, in 1666, the business of fire
insurance began to take practical shape, and in 1681 the regular fire insurance office was opened in London. The
first
first
office
in Scotland was established in 1720, the
many
in 1750, the first in the United States at Philadelphia
first
in Ger
1 Benjamin IFranklin as one of the directors. In course of time, fire insurance received some attention in In 1734 the first approach to modem English arithmetics. life insurance was made, but all members were rated alike,
in 1752, with
irrespective of age.
In 1807
we have
the
first
instance of
rating "according to age and other circumstances," It is interesting to observe that before the middle of tb& 1
DJS KOJKJJLN, Arith. Bootes, p. 69. "fesoraaee" in t&e Mnc&elop
* Axtiele
$& Ed.
MODERN
201
TISIE8
eighteenth century it was the custom in England to begin the legal year with the 25th of March, Not until 1752
did the counting of the
new year begin with the
first
of
1
January and according to the Gregorian calendar. In 1752, eleTen days were dropped between the 2d and 14th of
from "old
September, thereby changing
style" to
**new
style."
The order arithmetics
in
which the various subjects were treated in old
was anything but
The
logical.
definitions are
frequently given in a collection at the beginning.
I>ilwortb.
develops the rules for "whole numbers," then derelops the
same
and again for "decimal " rule of " three/' later the rule gives the " of three in fractions," and, again, the rule of three direct in rules for "rulgar fractions,"
fractions/*
Thus he
decimals." John Hill, in his arithmetic (edition of 1761) ? adds to this " the golden rule in logarithms." Fractions are taken
up
late.
Evidently
many
students had no expectation of ever
reaching fractions, and, for their benefit, the first part of the arithmetics embraced all the commercial rules. In the eigh teenth centery the practice of postponing fractions to the last became snore prevalent,* Moreover, the treatment of this subject
WBS usually vary meagre.
While the
better types
E. Srora in ys Nsw Mathematical Dictionary, London, 1743, speaks of the various early chaoses of the caAmsdar and then says, u Pope Gregory XTTI. pretended to reform it again, aad ordered his account *
to
&
current, as
it &tlll is
in all tiie
Rom*a
Cat^olick wnntriefi,"
Much
^ prejudice, no doubt, lay beneath the word pretended/' aiwi the word ** still," in this connection, now causes a ssaSe, * Jorn? KBRSET, in the l^th Ed, of WIMATS'S Aritkmt&ck, London^ 4t Far the Ease and Beoent of tlioee 1755, ^ys IB his preface, that desire only so much Skill in Arithmetic* oeeful
m
Trade, and such like ordinary Employments
;
m
the Doctrine of
(which, in the First Edition, was intermingled with Be&Mtes and Rules eQBC&rniQg Broken Numbers, commonly called Fraetioiifi) is now . So that J&&km&&& Wbole Nmnitea entirely hjuadkd apart . .
mm
m
A HISTORY OF MATHEMATICS
202
of arithmetics, "Wingate for instance, show how to find the L.CJD. in the addition of fractions, the majority of books take for the C.D. the product of the denominators. Thus,
Cocker gives 8000 as the C.D. of
+ f = 1A;
f, ,
^,
; Dilworth gives
Hatton gives + f = f} = *fIt was the universal custom to treat the rule of three under two distinct heads, " Eule of Three Direct " and " Eule
i
A
The former embraces problems like this, "If 4 Students spend 19 Pounds, how many Pounds will 8 students spend at the same Kate of Expence?" (Wingate,)
of Three Inverse."
The
inverse rule treats questions of this sort, "If 8 horses Days with a certain Quantity of Prov
will be maintained 12
How many Days will the same Quantity maintain In problems like the former the 16 Horses?" (Wijigate.) correct answer could be gotten by taking the three numbers in order as the first three terms of a proportion. In Wingate's
ender,
notation,
we would
have, If
Students
Pounds
Students
4
19
8.
But
in
the second example "you cannot say here in a direct propor tion (as before in the Eule of Three Direct) as 8 to 16, so is
12 to another
lumber which ought
to be in that Case as great
again as 12 ; but contrariwise by an inverted Proportion, begin ning with the last Term first as 16 is to 8, so is 12 to another ;
dumber."
This brings out very p. 57.) term of the "Eule of Three the clearly appropriateness all inverse" As problems were classified by authors under
two
(Wingate, 1735,
distinct heads, the direct rule
and the inverse
rule, the
pupil could get correct answers by a purely mechanical pro cess, without being worried by the question, under which rule is plainly and fully handled before any Ent'rance be made into die craggy Paths of Fractioos, at the Sight of which some Learners are so discouraged, that they make a stand, and cry out, nonplus ultra. There's
no Progress
farther."
MODERK TOO
203
Thus that part of the subject which
does the problem come ?
taxes to the utmost the skill of the
modem
teacher of propor No heed manner, tion, was formerly disposed of in an easy was paid to mental discipline. Nor did authors care what the pupil would do with a problem,
hand
to
what rule
when he was not
told before
it
belonged. with the time of Cocker, all demonstrations are Beginning
carefully omitted.
The only
proofs
known
to Dilworth are of
kind, "Multiplication and division prove each other/' The only evidence we could find that John Hill was aware
this
of the existence of such things as mathematical demonstrations lies in the following passage, " So i multiplied by % becomes \.
See this demonstrated in Mr. Leyburn's Cursus Mathematicus, What a contrast between this and our quotation on J
page 38/
from Tonstall
fractions
!
The
practice of referring to other
works was more common in arithmetics then than now. Cocker refers to Kersey's Appendix, to Wingate's Arithmetic, to Pitiscus's Trigonometric^
and to Oughtred's Claris for the
proof of the rule of Double Position. Cocker repeatedly gives on the margin Latin quotations from the Clams, from Alsted's
Jto&emoto, or GemmaFrisius's Mdkodus Fac&is (Witten berg, 1548).
Toward the end of the eighteenth century demonstrations begin to appear, in the better book% but th&y are oflea placed at the foot of the page, beneath a ItoraomtsLl line,, the rales and examples being above the
line,
By
this arrangement tfee
was appeased, while tibe teacher and pupil who did not care for proofs were least anaoyed by their
author's conscience
presence and could easily skip them. Moreover, the proofe and explanations were not adapted to the young mind ; there
was no abstract.
of objectteaching; the presentation was too True reform, both in England and America, began
trace
only with the introduction of Pestalozziaa ideas,
A HISTOEY OF MATHEMATICS
204
Mental arithmetic received no attention in England before the present century, but in Germany it was introduced during the second half of the eighteenth century. 1
Causes which Checked the Growth of Demonstrative Arithmetic in
England
Before the ^Reformation there was plished in the
way
little
or nothing accom
of public education in England.
In the
monasteries some instruction was given by monks, but we have no evidence that any branch of mathematics was taught to the In 1393 was established the celebrated " public
youth.*
school,"
named
Winchester,
and
in 1440, Eton.
In the six
eentory, on the suppression of the monasteries, The Merschools were founded in considerable numbers
teenth
ScJwcl, Christ Hospital, Rugby, Harrow, and which in England monopolize the title "public w and for centuries have served for the education schools
cktmt
Tatjftortf
others
In these of the sons mainly of the nobility and gentry. 8 schools the ancient classics were the almost exclusive subjects of study ; mathematical teaching was unknown there. Per haps the demands of everyday life forced upon the boys a
knowledge of counting and of the very simplest computations, but we are safe in saying that, before the close of the last century, the ordinary boy of England's famous public schools ,
s
p. 168.
Some Mea
of the state of arithmetical knowledge
may be gat&ere*! Shrewsbury, where a person was deemed f age when he knew .how to count up to twelve pence. (YearBooks Edw. L, XXL Ed. EOKWOOD, p. 220). See TTLOB'S Primitive from an ancient custom
Hew York,
1889,
Voi
at
I., p.
242.
Consult ISLIAC SHASPLBSS, Snglish Edwtxa&>n> H. y. RBDBAIX, 8cJt&oI&0$ Life in Merrie En^l^md^
Jos* TLMBS, SchoolDays of Eminent Mtn, London.
New Hew
York,
York, I8&1
;
MODSBH TOO
205
not divide 2021 by 43, though such problems had been performed centuries before according to the teaching of
Brahmagupta and Bhaskara bj boys brought up on the
farolf
has been reported that Charles XIL of Sweden considered a mam ignorant of arithmetic but half
banks of the Ganges,
a man. men.
It
Such was not the sentiment among English gentle Not only was arithmetic unstudied by them, but con
sidered beneath their notice.
If
we
are safe in following
Tiinbs's account of a book of 1622, entitled Peacha^'s Compl&xt
Gentleman, which enumerates subjects at that time among the becoming accomplishments of an English man of rank, then it appears that the elements of astronomy, geometry, aad
mechanics were studies beginning to demand a gentleman's remained untouched. 1 Listen
attention, while arithmetic still
to another writer,
Edmund
Wells.
In his Young Gentleman's
Course in 3athematicks 9 London, 1714, this able author aims to provide for gentlemanly education as opposed to thai of 2 w the meaner He expects those whom God part of mankind." has relieved from the necessity of working, to exercise their But must not " be so Brisk faculties to Ms
they
greater glory.
and Aiiy9 as to think, that the knowing how to cast Aocompt is requisite
only for sneit Underfill as Shopkeepers or Trades u IM>
men^ and if only for His sake of ^Mug care of themselves, thai gentleman ought to think ArUta^iek below "H not think an Estate below BJHL"
All the information
clo*s
we
could find respecting the education of the upper classes points to the conclusion that arithmetic was neglected, ant that Be
Morgan* was right in his statement that as Bale as the eigh teenth century there could have been no such tJii^g us a teacher of arithmetic in schools like Eton, In 1750, Warren Hastings, 1
TIMSS, 8c&&ol*Dv$*, p. 101.
* I>E
MORGAN,
Aritfi.
B&o&%
p.
61
206
A HISTORY OF MATHEMATICS
who had been
attending Westminster, was put into a commer he might study arithmetic and bookkeeping
cial school, that
before sailing for Bengal.
At
the universities little was done in mathematics before
the middle of the seventeenth century. It would seem that TonstalFs work was used at Cambridge about 1550, but in 1570, during the reign of given, excluding
all
Queen Elizabeth, fresh
statutes were
mathematics from the course of undergradu
presumably because this study pertained to practical life, and could, therefore, have no claim to attention in a university. 1 The commercial element in the arithmetics and algebras of
ates,
early times
Muhammed
was ibn
certainly very strong.
Musa and
Observe
how
the Arab
the Italian writers discourse on
questions of money, partnership, and legacies. Significant the fact that the earliest English algebra is dedicated by Becorde to the company of merchant adventurers trading to
is
Moscow. 2
Wright, in his English edition of Napier's loga " To the likewise rithms, appeals to the commercial classes :
Eight Honourable
And Eight
Worshipfvll Company Of Mer chants of London trading to the EastIndies, Samvel Wright wisheth all prosperitie in this life and happinesse in the life to
come." s
It is also significant that the first headmaster of the Merchant Taylors' School, a reformer with views in advance of his age, in a book of 1581 speaks of mathematical instruction,
thinks that a few of the most earnest and gifted students might hope to attain a knowledge of geometry and arithmetic from Euclid's Elements, but
fails to notice Recorded arithmetic,, which since 1540 had appeared in edition after edition. 4 Clas
1
2
and 8
BALL, Mathematics at Cambridge, p. 13. G. HEPFEL in 19th General Report (1893) of the A. 27.
NAPIER'S Construction (MACDOSAKD'S Ed.), p, 145. * G. HEPPBL, op. at, p. 28.
I.
G. T., pp. 26
2G7
M0DEJ&N TIMIS
men were
eyidently not in touch with the that of tima mathematics sical
new
in the
This scorn and ignorance of the art of computation by all but commercial classes is seen in Germany as well as England. Kastner 3 speaks of it in connection with German Latm 2
linger refers to it repeatedly. It was not before the present century that arithmetic and
schools;
other branches of mathematics found admission into England's
At Harrow "vulgar fractions, Euclid, geog " 8 At in 1829. and modern history were first studied raphy, the Merchant Taylors' School "mathematics, writing, and arithmetic were added in 1829." 4 At Eton mathematics was public schools.
**
not compulsory
till
1851." of the
clusive classicism
fi
The movement against the ex was led by Dr. Thomas Dr. Arnold of Matthew Arnold.
schools
Arnold of Bugby, the father
favoured the introduction of mathematics, science, history, and politics.
Since the art of calculation was no more considered a part of a liberal education than was the art of shoemaking, it is natural to find the study of arithmetic relegated to the com mercial schools. The poor boy sometimes studied it ; tie rich
boy did not seed it
In Latin schools
it
was unknown, but in
schools for the poor it was sometimes taught ; for example, in a " free grammar school founcbd by a grocer of IxHkba in
1553 for thirty *of the poorest men's sons* of Guiiford, to be taught to read and write English and cast asocomnts perfectly, That a so that they should be fitted for apprentice."* science, ignored as
a mental
ment,
is
not strange.
See pp.
to receive fuller develop It was, in fact, very natural that it fail
ffl., p. 429.
*
TiaiBS, p. 84.
5, 24, 112, 140, 144.
*
SsAmpLESS, p. 144. TIMB&, p. 88,
,
8
and studied merely as
discipline,
an aid to material gain, should
p,
m.
*
A HISTORY OF MATHEMATICS
208
should entirely discard those features belonging more prop erly to a science and assume the form of an art. So arithmetic
reduced
to a
itself
mere collection of
rules.
The
ancient
Carthaginians, like the English, studied arithmetic, bnt did
not develop it as a science. " It is a beautiful testimony to the quality of the Greek mind that Plato and others assign as
a cause of the low state of arithmetic and mathematics among . . the want of free and disinterested inves the Phoenicians .
l
tigation."
It will be observed that during the period under considera tion, the best
English mathematical minds did not make their The best intellects held aloof
influence felt in arithmetic.
from the elementary teacher; arithmetical texts were written by men of limited education. During the seventeenth and eighteenth centuries England had no Tonstall and Becorde,
and Regiomontanus, no Pacioli and Tartaglia, to her arithmetical books. To be sure, she had her compose her Kewton, Cotes, Hook, Taylor, Maclaurin, De "Wallis, BO
Stifel
Moivre, but they wrote no books for elementary schools ; their influence on arithmetical teaching was naught. Contrast this period with the present century. Think of the pains taken by Augustus De Morgan to reform elementary mathematical The man who could write a brilliant work on instruction.
the Calculus,
who could make new
discoveries in advanced
logic, was the man who translated algebra, series, Bourdon's arithmetic from the French, composed an arith metic and elementary algebra for younger students, and
and
in
endeavoured to simplify, without loss of rigour, the Euclidean " geometry. Again, run over the list of members of the Asso
Improvement of Geometrical Teaching," and of the Committee of the British Association on geometrical
ciation for the
*
J.
K, F. ROSBNKRJLSFZ, T,
Hew Yodc,
PMl&so^ vf E^&s^on^
1888, p. 215,
translated fey A. G.
M0DEBH TQCKS teaching, said
you
will find in
it
209
England's most brilliant
mathematicians of our time.
The
imperfect interchange of
ideas between writers on
advaaced and those on elementary subjects is exhibited in the mathematical works of John Ward of Chester. He pub
Compendium of Algebra, and in 1706 his Young Matkemctfidam's Guide. The latter work appeared in the 12th edition in 1771, was widely read in Great Britain, and well approved in the universities of England, Scotland, It was once a favourite textbook in Ameri and Ireland, can colleges, being used as early as 1737 at Harvard College, and as late as 1787 at Yale and Dartmouth. In 475 pages, lished in 1695 a
the book covered the subjects of Arithmetic, Algebra, Geom etry, Conic Sections, and Arithmetic of Infinites, giving, of
mere rudiments of each, to raise a binomial to positive integral "without the trouble of continued involution" and powers remarks that when he published this method in his Compaq
course, the
Ward shows how
dium of Algebra he thought that he was the it,
but that
sines
first
inventor of
he has found in Wallis's History of Algebra
karaed Sir Isaac Newton had discovered it long took a quarter el a ceatary for the news of binomial Newton's More fMsewerf t seadh Join Ward. ovir over, it looks very odd t* se m. Ward's Guide of 1771 a soft f a century after Kewton^s diseowrj el iLwxwm integral calculus, such as was empkjj ed Walli^, Oaralieo* Fermat, and Boberval befwe the mvesmtkm f flvxiaias. It is difficult to discover a time when in j eivilid fej advanced and elementary writers cm mathematics were more thorou^ilj out of touch with each other than in Englaaid dur We earn ing the two centuries and a half preceding 1800. think of no other instance in science in which the failure of the best minds to influence the average has led to snefe "that
ifafi
before.**
It
%
A HISTORY OF MATHEMATICS
210
In recent times we have heard it deplored that in some countries practical chemistry does not avail itself Fears have been of the results of theoretical chemistry. lamentable results.
expressed of a coming schism between applied higher mathe But thus far matics and theoretical higher mathematics. these evils appear insignificant, as compared with the wide
spread repression and destruction of demonstrative arithmetic, arising from the failure of higher minds to guide the rank and file. Neglected by the great thinkers of the day, scorned
by the people of rank, urged onward by considerations of purely material gain, arithmetical writers in England (as also in Germany and France) were led into a course which for centuries blotted the pages of educational history. In those days English and German boys often prepared for a business career by attending schools for writing and arithmetic. During the Middle Ages and also long after the
invention of printing, the art of writing was held in high esteem. In writing schools much attention was *given to The schools embracing both subjects are fancy writing.
always named schools for "writing and arithmetic," never " " arithmetic and writingwriting." The teacher was called master and arithmetician."
Cocker was skilful with the pen, and wrote many more books on calligraphy than on arithmetic. As to the quality of the teachers, Peacham in Ms Com$ea Gentleman (1622) stigmatizes the schoolmasters in his own day as rough and even barbarous to their pupils. Domestic tutors he represents as still worse. 1 The following from
an arithmetic of William Webster, London, 1740, gives as idea of the origin of
Man
has tried
many commercial
all Shifts,
and
schools.
still failed, if
2
oat anything like a fair Character, tho never so 1
TIMBS,
p.
101
a
DB MOKGAK,
"When
a
he can but scratch stiff
Ariih. Books, p. d$.
and
211
MODERN TIMES unnatural and
lias
got but Artihm&tick
enough in
Ms Head
to
a Mile, he compute the Minutes in a Year, or the Inches in makes his last Recourse to a Garret, and with the Painter's Help, sets up for a Teacher of Writing and Arithmetic^; where, of by the Bait of low Prices, he perhaps gathers a Number Scholars." there were
No
doubt in previous centuries, as in all times? teachers, but the large mass of school
some good
masters in England, Germany, and the American colonies were Some of the more of the type described in the above extract ambitious and successful of these teachers wrote the arithmetics for the schools.
No wonder
that arithmetical authorship and
teaching were at a low ebb. To summarize, the causes which checked the growth of
demonstrative arithmetic are as follows
:
not studied for its own sake, nor valued (1) Arithmetic was for the mental discipline which it affords, and was, consequently,
learned only by the commercial classes, because of the material gain derived from a knowledge of arithmetical rules. (2)
The
best minds failed to influence
and guide the average
minds in arithmetical authorship.
m
Ari&met*e0Z
A
great reform in elementary education was initiated in Switzerland and Germany at the begimmiiig of the nine
teenth century.
Pestalozzi emphasized in all iBstraetiOT
tlfee
"Of
the instruction at Yrarchm, necessity of objectteaching, the the most successful in opinion of those who visited the school,
was the
instruction in arithmetic.
The
children a^e
described as performing with great rapidity very difficult tasks Pestalozzi based his method here, as in in headcalculation, other subjects, on the principle that the individual should be similar to that which brought to the knowledge by the road
A HISTORY OF MATHEMATICS
212
the whole race liad used In founding the science.
Actual
counting of things preceded the first Cocker, as actual meas The child urement of land preceded the original Euclid. out the find to and to count various must be taught things processes experimentally in the concrete, before he is given any abstract rule, or is put to abstract exercises. This plan is
now adopted
in
German
schools,
and many ingenious con
trivances have been introduced by which the combinations of 1 Much things can be presented to the children's sight."
remained to be done by the followers of Pestalozzi in the way of practical realization of his ideas. Moreover, a readjust
ment of the course
was needed, for Pestalozzi arithmetic which are applied in
of instruction
quite ignored those parts of
Until about 1840 Pestalozzi *s notions were everyday life. followed more or less closely. In 1&42 appeared A. W. Grube*s Leitfaden,
which was based on
Pestalozzi's
idea of object
teaching, but, instead of taking up addition, subtraction, mul tiplication, and division in the order as here given, advocated
the exclusion of the larger numbers at the start, and the teach ing of all four processes in connection with the first circle of numbers (say, the numbers 1 to 10) before proceeding to For a time Grube's method was tried in a larger circle.
Grermany, but
it
soon met with determined opposition.
The
experience of both teachers and students seems to be that, t reach satisfactory results, an extraordioarf expenditure of
energy
is
demanded.
method seems
In the language of physics, Grabefo low efficiency. 2
to be an engine having a
Into conservative England Pestalozzian ideas found tardy ad1
E. H, QCICK, Educational Eeformers, 1879,
s
For Hie history of
p. 101.
arithmefcieal teaching is
Germany during Hie
For a discussion eea&ary, see USGEE, pf*. 175233. psycttoiogical bearing of Gmbe's method, see MC!/ELLAN AJTB Psychology of Number, 1396,
p.
SO
ei seg.
fcbe
213
MODEEK TIKIS mittanea
At the time when De Morgan began to write
(alxrat
the level 1830) arithmetical teaching had not risen far above In more recent time the teach of the eighteenth century. ing of arithmetic has been a subject of discussion by the Association for the ImproTement of Geometrical Teaching." Among the subjects under consideration have been the mul **
tiplication
l
and division of concrete quantities, the approxi
mate multiplication of decimals (by leaving off the tail of the product), and modification in the processes of subtraction, mul The new modes of subtraction and tiplication, Mid division. Austrian methods," be division are called in Germany the ft
cause the Austrians were the
first to
In England
adopt them.
mode of division goes by the name of " Italian method." 2 The ''Austrian" method of subtraction is simply this: It
this
shall be performed in the
same way as
u
change
"
is
given in
a store by adding from the lower to the higher instead of passing from the higher to the lower by mental subtraction. " nine and semn are Thus, in 76 49, say sixteen, five and
The
two are seven."
practice of adding 1 to the 4, instead
was quite common during the Renais by Maximus Planudes and In Adam Riese's works Aiiam Biesa Pemerfeadi, Georg there is also an approach to the determination of the differ
of subtracting 1 sance.
fitjta 7,
It wias used, for instance,
ence by figuring from the subtrahend upwawte, In the above from 10 and add the remainder example, he would subtract 1 to the minuend
6,
thus obtaining tLe answer ?*
A
recent
American textbook on algebra explains subtraction on the u Austrian principle of the *
See the Geaervl Bqp&rtp sine 1888, specially *liJe for 1882 said name 4t Italian BaettKxT 1 was traced by Mr. Laagley back to an of 17. See General Rsport of A. L G. T., IWi, p. $4. antitoefcle English * Tins
ttwl hi&oriscber
BdewMmtog, Konigsberg,
1892, p. 13.
A HISTORY OF MATHEMATICS
214
In multiplication, the recommendation is to begin, as in alge bra, with the figure of highest denomination in the multiplier.
The
great advantage of this shows itself in decimal multiplica we desire only an approximate answer correct to,
tion in case
say, three or five decimal places.
We have seen that this method
was much used by the Florentines and was called by Pacioli by the little castle." Of the various multiplication processes
"
of the Renaissance, the fittest failed to survive.
1
The
process
2
now advocated was
taught by Nicholas Pike in the following " It is : example required to multiply 56.7534916 by 5.376928, and to retain only five places of decimals in the product."
The "Austrian" 56.7634910
.
method
^vision simply calls for the multiphcation of the divisor and the subtraction *
82 9673.6
28373740 1702605 397274 8*062
or "Italian"
,
.
.
.
.
....
.....
from the dividend simultaneously, so only the remainder is written
that
down on It
113
See our illustration, paper. doubtful whether this method is
is
preferable to our old method, except
45
for
305.15&43
naturally
978)272862(279
,
.
K?" l^j
8802
The
a^^
We computers. saves computor paper *
rapid
fear that the slow
^ e exPense _
" Austrian
*
i
of mental energy. method of division " is 
" But 1 This process was favoured by Lagrange. He says nothing compels us to begin with the right side of the multiplier ; we may as well begin with the left side, and I truly fail to see why this method is not preferred, for it has the great advantage of giving the places of highest :
value first ; in the multiplication of large numbers we are often interested most in the highest places." See H. NiEDEEMtJLLEx, Lagrang&s Jf&i&eDeutsche Separatausgabe, Leipzig, m<sicke Elemerdarcorlesungen. This publication, giving five lectures on arithmetic and 1880, p. 23. algebra, delivered by the great Lagrange at the Normal School in Paris, ia 179&, 2
is
very interesting for several reasons.
N Pixm, dr&kmetic, abridged for the use of sc&ools, 3d Ed., Worees
te, 1798,
p.
9&
MODERN TIMES
215
not entirely new; in the "galley" or "scratch" method, the partial products were not written down, but at once subtracted,
and only the remainder noted. In principle, the "Austrian method" was practised by the Hindus, of whose process the 1 " scratch " method is a graphical representation.
Arithmetic in the United States
Weights and Measures.
The weights and measures
intro
duced into the United States were derived from the English. "It was from the standards of the exchequer that all the weights and measures of the United States were derived, until Louisiana at first recognized Congress fixed the standard. standards derived from the French, but in 1814 the United The States revenue standards were established by law." 2 actual standards used in the several states
and
in the
custom
houses were, however, found to be very inaccurate. In the construction of accurate standards for American use, our Gov
ernment engaged the services of Ferdinand Rudolph Hassler, a Swiss by birth and training, and a skilful experimental 3 The work of actual construction was begun in 1835. 4 ist.
In 1836, carefully constructed standards were distributed to the customhouses and furnished the means of uniformity in the collection of the customs. Moreover, accurate stand ards were distributed by
the
general Government to
the
For further information regarding the "Austrian" method of suband division, see SADOWSKI, op. cit. ; UNGER, pp. 2132182 Eeport of the Secretary of the Treasury on the Construction and Dis Ex. Doc. No. 27, 1857, p. 36. tribution of Weights and Measures. 8 Consult the translation from the German of Memoirs of Ferdinand 1
traction
Rudolph Hassler, by EMIL ZSCHOKKE, published in Aarau, Switzerland, See also 1877 with supplementary documents, published at Nice, 1882. Teach, and Hist, of Math, in the U. &, pp. 286289. 4 Ex. Doc. No. 84 (Report by A. D. Bache for 184647), p. 2. ;
A HISTOEY OF MATHEMATICS
216
various States, with
tlie
view of securing greater uniformity,
It was recommended that each State prepare for them a the charge of some fireproof room and place them "under use and safe their to scientific person who would attend
keeping."
In 1866 Congress authorized the use of the Metric System United States, but unfortunately stopped here and allowed the nations of Continental Europe to advance far in the
beyond us by their adoption of the Metric System
to the exclu
sion of all older systems.
In the currency of the American colonies there existed great " At the time of the adoption of our diversity and confusion. decimal currency by Congress, in 1786, the colonial currency or bflls of credit, issued by the colonies, had depreciated in value, this depreciation, being unequal in the different colonies, 1 gave rise to the different values of the State currencies."
and
"
our early arithmetics were " practical " reduction of metics, they, of course, gave rules for the
Inasmuch as
all
aritbcoin.**
Thus Pike's Arithmetic* devotes twentytwo pages to tli statement and illustration of rules for reducing "Itfewhampand Virginia "Newyork and
saire, Massachusetts, Ehodeisland, Connecticut,
currency" (1) to
"Federal Money,"
(2) to
Northcarolina currency," (3) to "Pennsylvania, Newjersey, " Irish ." money, Delaware, and Maryland currency, (6) to " Canada and Kovaseotia currency^ (8) to "Livres To^ur(7) to 5*
.
.
" nois," (9) to Spanish milled dollars."
reducing Federal
Money
to
Then follow rules for ^Newengland and Virginia cur
J It is easy to see how a large share of the pupil's rency / etc. time was absorbed in the mastery of these roles. The ters on reduction of coins, on duodecimals, alligation,
*
*
BOBIN&ON, Progressive Higher Arithmetic, 1874, p. 190. The and Complete System of ArmmeUc, abridged for *&e use
Nm
,
1798, Wortsesier, pp. 117130.
MODEEK TIMBS give evidence of the
217
homage that Education was forced
to
pay to Practical Life, at the sacrifice of mailer better fitted
mind of youth. With a Yiew of supplying the Information needed by merchants in business, arithmetics discussed such subjects as the United States Securities, the to develop the
canons
rules
adopted by
the United States, and by the
State governments on partial payments. Auikors and BOG&&. The first arithmetics used
American
the
in
were English works Cocker, Rodder, 1 Fisher" (Mrs. Slack), Daniel Fenning. Dilworth, "George colonies
:
The
earliest arithmetic written and printed in America ap peared anonymously in Boston in 1729. Though a work of considerable merit, it seems to Lave been used very little in 5
early records
we have found no
reference to
later, at the publication of Pike's Arithmetic,
fifty years the former work it;
was completely forgotten, and Pike's was declared to be the American arithmetic. Of the 1T29 publication there axe two copies in the Harvard Library and one in the Congres earliest
rapky
its
1
In Appleton's Cydop
sional Library.
wood, than pnxf S80r of mafeeniatics &fc Harvard College, but on ih0 titlepagp of one f the optas in the Haarvard Library, is
written the following: "iSupposed that Sam} Greenwood thereof, by others said to be by Isaac Green
was the author wood,"
IB 1788 appeared at Newbnryport the
CkmiMe
System of Arithmetic try
New
W^AsAm Pike
a graduate of Harvard College.* It was intended for adva&eed schools, and contained, besides the ordinary subjects of that time, logarithms, trigonoinefery, algebra, and aosiiie seeiaeiis; but these latter subjects were so briefly treated as to possess In the ** Abridgment for the Use of Schools/* little Yalue. i
*
See Teack. a*d Ibidem, p. 14.
m&.
of
Jf&,
in tk *
U.S., pp. 1216,
B&lem, pp.
4&,
4&
A HISTORY OF MATHEMATICS
218
which was brought out
at Worcester in 1793, the larger work " in the of preface as now used as a classical book in spoken all the Newengland Universities." recent writer 1 makes is
A
Pike
responsible for all
the abuses in arithmetical teaching To us this con
that prevailed in early American schools. demnation of Pike seems wholly unjust
even
if
we admit
It is unmerited, that Pike was in no sense a reformer among
arithmetical authors. Most of the evils in question have a far remoter origin than the time of Pike, Our author is fully up to the standard of English works of that date. He can no more be blamed by us for giving the aliquot parts of pounds
and
u tare shillings, for stating rules for
and trett," for dis u cussing the reduction of coins/' than the future historian can blame works of the present time for treating of such atrocious areiaiaoas as tkafc 3 ft = 1 yd., 5 yds. = 1 L, 30 sq. yds. = 1 So long as this free and independent people down to such relics of barbarism, the arith
sq. rtl, etc.
to be tied
cannot do otherwise than supply the means of acquir the ing precious knowledge, At the beginning of the nineteenth century there were three ** ?? great arithmeticians in the United States Nicholas Pike, Daniel Adams, and Nathan Daboll. The arithmetics of Adams :
(1801) and of Daboll (1800) paid more attention than that of
Pike to Federal Money.
Peter Parley tells us that in conse quence of the general use, for over a century, of Dilworth in
American schools, pounds, shillings, and pence were classical, and dollars and cents vulgar for several succeeding generations. U I would not I would not gire a penny for it" was genteel; " was plebeian. Eeform in arithmetical teaching in the United States did
give a cent for
B0fc *
it
begin until the publication by Warren Colburn, in 1821,
GBORGB H. MAHTIX, The Evolution of p. 10SL
tht Massachusetts Public
HOBBBK TIMES of the Intdkctttcd
Armm&k*
This was
219 tlie
first
fruit of
Pestalozzian ideas on American
Like Pestalozzi, Colsoil. horn's great success lay in the treatment of mental arithmetic. The success of this little book was extraordinary. Bat Ameri
can teachers in Colburn's time, and long
after,
nerer quit
succeeded in successfully engrafting Pestalozzian principles on written arithmetic. Too much time was assigned to arith
There was too little objectteaching ; either abstruse reasoning, 2 or no reasoning at all ; too little attention to the art of rapid and accurate computation; too metic in schools. too
much
much
attention to the technicalities of commercial arithmetic.
During the
last ten
years, however, desirable
reforms hare
been introduced. 8
"Pleasant and Diverting Questions"
In English and American editions of Dilworth, as also in j Daniel Adams's 5efto/ar s Arithmetic* we find a curious col lection of "Pleasant
and diTerting questions."
We
haTe
all
heard of the farmer, who, haying a fox, a goose, and a peck of corn^ wished to cross a river; but, being able to carry only one at a time, was confounded as to how he should take " WARmBi? CoLB0En*s Firxt Lessms featB bees abused by being put hands of eiiildren too early, and has been productive of utmost as modi tana as good." RT. I^EIOMAS BfeuE^ %%$ IVwe Or&er $f Sswtffas, 1
in the
1876, p. 42. *
u The teacher wbo has been accustomed to the modern erroneous
metbod of teaching a
child to reason out his processes from the beginning method of gaining facility IB the operations, before attempting to explain them, fa the method of N&tare and that It is m& onl y much pleasanter to the child, but thai it will make & better mathe T. HIIX, op. e&., X 45. matician of him." * For a more detailed history of arithmetical teaching, see Teach, and
may be
assured, this
;
Hist, of *
Jtok
in tkf U.S.
Sereatk M,,
Mcm^fe, Vt,
1812, p,
A HISTORY OF MATHEMATICS
220
them
across so that the fox should not devour the goose, nor
Who has not been entertained by the the goose the corn. husbands with their wives may three how jealous problem, cross a river in a boat holding only two, so that none of the three wives shall be found in
company
of one or
two men
Who
has not attempted to in that so a nine any three figures in a square digits place unless her husband be present ?
line
may make
pected the creatures of
just
great
15?
N"one of us, perhaps, at first sus
antiquity of these apparently newborn Some of these puzzles are taken by
fancy.
Dilworth from Kersey's edition of Wingate. Kersey refers the reader to " the most ingenious Gaspar Backet de M&ziria ?'
PrMZmes pZaimnts et d&ectables qui sefont
in his little book, fe
jwmm,
in the modified version of the wolf, goat,
and cabbage
and their wives were puzzle, The three jealous husbands the same question with known to Tartaglia, who also proposes 1 four husbands and four wives,
We take these
to be modified
and improved versions of the first problem. The three jealous husbands have been traced back to a MS. of the thirteenth century, which represents two German youths, Jirri and 1 The MS. contains proposing problems to each other. " There were three brothers in also the following: Firri says:
Tym,
Cologne, having nine vessels of wine. The first Tessel con tained one quart (amam), the second 2, the third 3, tfa fourth 4, the fifth 5, the sixth 6, the seventh 7, the eight!*
Divide the wine equally among the three without mixing the contents of the vessels." This
B7 ^be ninth
9.
PSULOOCK, p. 47S.
question since
it
is
MODEEH TIMES
221
closely related to the third
problem given above,
gives rise to the following magic square
demanded by
that problem.
Magic squares were known to the Ara&s and, perhaps, to the Hindus. To the Byzantine writer, Mosehopulns, who lived in Constantinople in the early part of the fifteenth century, appears to be due the introduction into Europe of these curious and ingenious products of mathematical thought.
believed
them
Mediaeval astrologers
to possess mystical properties
and
when engraved on silver plate to be a charm against plague. 1 The first complete magic square which has been discovered in the Occident is that of the German painter, Albrecht Dtoer, found on his celebrated woodengraving, " Melancholia."
Of
interest is
the following problem, given in Kersey's
Wingate: "15 Christians and 15 Turks, being at sea in one and the same ship in a terrible storm, and the pilot declaring a necessity of casting the one half of those persons into tha sea, that the rest might be saved; they all agreed, that the persons to be east away should be set out by lot after this
manner, "rix. fee 3 persons should be placed in a round form like a ring, a^el t&an beginning to wont at one of the pas sengers, and proceeding cifenlarly, vwy ninth person should be east into the sea, until of the 30 persons there remained
The question is, l*w H&os 30 persons ought to be the lot might infallibly that upon the 15 Turks placed, and not upon any of the 15 Cari&tijwst** EJerawey leis ti& only 15.
Ml
letters a, e,
i,
gives the verse
o,
&
stand, respectively, for 1,
From numbers' Never
i
aid
and
2#
%
4^
art,
wm fame depart.
For the history of Magic Squares, see GBKTHJSR, Vcrmisckte is in &e artkle " OL IV. TMr
awdtoj^efc,
Squares" in
tlteory
JOHSFSOJI'S Universal
developed
A HISTOET OF MATHEMATICS
222
Tlie rowels in these lines,
nately the
taken in order, indicate alterto be placed to
number of Christians and Turks
=
=
4 Christians, then w 5 Turks, then gether; i.e., take e 2 Christians, etc. Bachet de Meziriac, Tartaglia, and Cardan give each different verses to represent the rule. Ac 1 cording to a story related by Hegesippus, the famous historian Josephus, the Jew, while in a cave with 40 of his country men, who had fled from the conquering Eomans at the siege
Ms life by an artifice like the above. Eather than be taken prisoners, his countrymen resolved to kill one another* Josephus prevailed upon them to proceed of Jotapata, preserved
by lot and managed Both agreed to lire.
The problem of
it
so that he and one companion remained.
the 15 Christians and 15 Turks has been
by Cardan Lmdus Josqpk, or Joseph's Play. been found in a French work of 1484 written by
Called
Ch^nei* and eeatories.*
MSS.
in
Daniel
of the twelfth, eleventh,
Adams
It has JSTieolas
and tenth
gives in his arithmetic the follow
ing stanza;
"As I was going to St.
Ives,
I suet seven wives.
Every wife Iiad seven sacks; Every sack had seven eats ; Every cat had seven kite Kits, cats, sacks, and wives, How many were going to St. Ives? M :
Compare tMs with
Fibonaci's " Seven old
women go to Borne," sad with the problem in the Ahmes papyrus, and we perceive that of all problems in "mathematical recreations"
etc.,
this is the oldest
Pleasant and diverting questions were introduced into some English arithmetics of the latter part of the seventeenth and etc.,
HL, Ch. t
IS.
2
CAITTOR, Yol.
18P4, p, 116 Mi
II., p.
302.
MODEEK TIMES of the eighteenth centuries.
223
In Germany ibis subject found
entrance into arithmetic during the sixteenth century. Its In the seven to make arithmetic more attractive.
aim was
teenth century a considerable
number
of
German books were
1
wholly devoted to this subject Certain writers have found amusement in speculations over the origin of the $. Over a dozen different theories have been advanced, but no serious effort was made to verify any of them. A careful study of manuscripts, made recently 32 has shown that the $ is the lineal descendant of the Spanish abbreviation p* for "pesos," that the change from the rescent
p
s
to $
flo
was made about 1775 by EnglishAmericans
who came in business relations with Spanish Americans. The earliest known printed $ occurs in an arithmetic, Chauncey Lee's American Accomptant, published in 1797 at Lansing
burgh. 1 2
See ^. CAJOEI in the Popni&r Science Mon&tfr December, Bdmce, H. ft., Vol 38, 1913, pp. 848860*
pp. 621630;
ALGEBRA
The Renaissance
ONE
of the great steps in the development of algehra during sixteenth the century was the algebraic solution of cubic The honour of this remarkable feat belongs to equations.
the Italians. 1
The made by
attack upon cubic equar tions "was Scipio Ferro (died in 1526), professor He solved cubics of the form of mathematics at Bologna. a? 4
mx = n,
first successful
but nothing more
is
known
of his solution than
It was the that he taught it to his pupil Floridus in 1505. and centuries those for in afterwards during days practice
teachers to keep secret their discoveries or their new methods of treatment, in order that pupils might not acquire this knowledge, except at their own schools, or in order to secure
an advantage over rival mathematicians by proposing prob lems beyond their reach. This practice gave rise to many One of the most disputes on the priority of inventions.
famous of these quarrels arose in connection with the dis In 1530 covery of cubics, between Tartaglia and Cardan. one Colla proposed to Tartaglia several problems, one leading to the equation
a?+ px? = q.
method of resolving 1
this,
Tfoe geometric solution
The latter found an imperfect made known his success, but kept
had been given previously by the Aiaiw. 224
THE REHAISSAKCB
Ms
solution secret
225
This led Ferro's pupil Floridus to pro
=
how to solve s? 4 m& n. Tartaglia challenged him to a public contest to take place Feb* 22, 1535. Meanwhile he worked hard, attempting to solve other cases claim his knowledge of
of cnbic equations, and finally succeeded, ten days before the 8 mx j n. At the appointed date, in mastering the case x
=
man
The one who should proposed 30 problems. be able to solve the greater number within fifty days was to
contest each
Tartaglia solved his rival's problems in two could not solve any of Tartaglia's. Thence Floridus hours; forth Tartaglia studied cubic equations with a will, and in 1541 he was in possession of a general solution. His fame
be the
victor.
began to spread throughout
what
It is curious to see
Italy,
interest the enlightened public took in contests of this sort
A
mathematician was honoured and admired for his
ability.
Tartaglia declined to make known his method, for it was his to write a large work on algebra, of which the solution
But a scholar
of cubics should be the crowning feature.
Milan, named Hierommo Cardano
(15011576), after
of
many
and the most solemn promises of secrecy, suc ceeded la obtaining from Tartaglia the method. thereupon inserlsed it in a matiieiBatteal work, the Ars then in preparation, which lie published in 1545. This breach of promise almost drove Tartaglia mad. His first step was solicitations
to write a history of his invent!!^ boi t late Cardan,
contest.
he challenged him and
Tartaglia excelled in
@
Ms
pupil Wmm&. to a of solving pr^ila% pcrwer final 0nte03ae of all Has was
Ms
bat was treated unfairly. The that the man to whom we owe the chief algel^a
made
in the sixteenth century
discovery in question went by the
wa&
name
ontritmfck
forgotten,
to
aad the
of Cardan's solution.
Cardan was a good mathematician, but the association of Ms name with the discovery of the solution of cubks is a Q
A HISTORY OF MATHEMATICS
226
gross historical error
and a great
injustice to the genius of
Tartaglia.
The
success in resolving cubics incited mathematicians to
extraordinary
efforts
toward the solution of
equations of
The
solution of equations of the fourth degree Mgher degrees. was effected by Cardan's pupil, Lodovico Ferrari. Cardan had the pleasure of publishing this brilliant discovery in the Ars Magna of 1545. Ferrari's solution is sometimes ascribed to Bombelli,
who
is
no more the discoverer of
it
than Cardan
of the solution called by his name. For the next three centuries algebraists made innumerable attempts to discover algebraic solutions of equations of higher degree than the is
fourth. It is probably no great exaggeration to say that every ambitions young mathematician sooner or later tried his skill in this direction. At last the suspicion arose that this
problem, like the ancient ones of the quadrature of the circle, duplication of the cube, and trisection of an angle, did not
admit of the kind of solution sought. To be sure, particu forms of equations of higher degrees could be solved satis
lar
For instance, if the coefficients are all numbers, some method like that of Vieta, Newton, or Horner, always factorily.
enables the computer to approximate to the numerical values But suppose the coefficients are letters which
of the roots.
may
stand for any number whatever, and that no relation is to exist between these coefficients, then the problem
assumed
assumes more formidable aspects. Finally it occurred to a few mathematicians that it might be worth while to try to prove the impossibility of solving the quintie algebraically; is, by radicals. Thus, an Italian physician, Paolo Ruffini
that
1 (17651822), printed proofs of their insolvability, but these proofs were declared inconclusive by his countryman MalfattL 1
See H. BuKKHAEixr, " Die An&nge der Grappentbeorie mad Paolo ZaJscAr./. Math. u. fkysik, SuppL, 1892.
Rafci"
m
THE BKHAISSAHOE
227
Later a brilliant young [Norwegian, Nids Henrys Abd (18021829), succeeded in establisMng bj rigorous proof that the general algebraic equation of the fifth or of higher degrees
cannot be solved by radicals. 1
A
modification of Abel's proof
was given by WantzeL* Eeturning to the Benaissanee it is interesting to ob&erre, that Cardan in his works takes notice of negative roots of an equation (calling them fiditious, while the positive roots are called real),
and discovers
all
three roots of certain numerical
(no more than two roots having ever before been found in any equation). While in his earlier writings he rejects imaginary roots as impossible, in the Ars Magna he cubics
exhibits great boldness of thought in solving the problem, to divide 10 into two parts whose product is 40, by finding the
answers 5{V
15 and 5
V
15,
and then multiplying
8 Here for the first together, obtaining 25 h 15 = 40. time we see a decided advance on the position taken by the
them
Hindus.
Advanced views on imaginaries were held
also
by
who
published in 1572 an alge bra in which he recognized that the socalled irreducible case in cuMcs gives real values for the roots. Bapfoael
BomMH,
of Bologna,
It may be instructive to give examples of the algebraic notation adopted in Italy in those days. 4 1
See CSBLLB'S Journal, I., Waatzel's proof, translated from Sera&'s 0@*sr$ &M$$MK Bagfatemv, was published IB VoL IV., p. 65, of ^e A&$$a&, edited &y JOWL K. HxDB.ICKB of Des Homes. While tfee quintic eamaofc be s&lfed by radicals, a *
transcendental solution, invoking elliptic integrals, was given by Hermite (in the C0sp*. Send., 1858, 1866, 18$@) and fej Kitweefcer la 186& translation of the solution by elliptic integrals, taken from Biiot and
A
Bouquet's Tlieory of Elliptic Functions,
AM&$&,
Vol. V., p. 161.
is
likewise paWislied in the
s
CANTOR, IL, 506. ^8 ; tfae Tatoa of x givea om
* CAKTO&, IL, 320 ; HiTTHiB^EN, p. following page Is the solution of the caMe ia tte prerioos u yn or " ** is a sign of aggregation or jolai root.
&
A HISTORY OF MATHEMATICS
228 Pacwli
Cardan
:
:
5.
The
V40  V320.
$ V 40 m $ 320, Cubus p 6 rebus v. cu.
.
= 20, m 5. v. cu. 3, 108 m 10, + 10  V/V108  10.
aequalis 20,
108 p. 10

$
f
6x
Italians were in the habit of calling the
quantity cosa, "thing." In as early as the time of John
Germany
this
Widmann
as a
unknown word was adopted
name
for algebra:
he speaks of the "Begel Algobre oder Cosse"; in England this
new name
English work
for algebra, the cossic art, gave to the
thereon,
first
by Robert Keeorde, its punning title The Germans made truly a cos ingenii.
the Whetstone of Witte, and important contributions to algebraic notation. The signs, mentioned by us in the history of arithmetic, were, of course, introduced into algebra, but they did not pass into gen " It is eral use before the time of Vieta, very singular," says "that discoveries of the HaJlam, greatest convenience, and,
+
apparently, not above the ingenuity of a village schoolmaster, should have been overlooked by men of extraordinary acuteness like Tartaglia, Cardan, and Ferrari and hardly less so that, by ;
dint of that acuteness, they dispensed with the aid of these contrivances in which we suppose that so much of ths utility of algebraic expression consists." Another important symbol introduced by the Germans is the radical sign. In a manu script published some time in the fifteenth century, a dot
placed before a number is made to signify the extraction of a root of that number. Ghristoff Rudolf, who wrote the earliest textbook on algebra in the German language (printed 1525), remarks that "the radix quadrata is, for brevity, designated in Ms algorithm with the character 4*" Here the dot y, as found in the manuscript has grown into a symbol much like our own. With him VVV and VV stand for the cube aad
V
THE BENAISSASTCE the fourth roots.
The symbol
V was
229
*****
by Michael
Stifel
edition of (1486 ?1567), vho, in 1553, brought out a second Budolff s Coss, containing rules for solving cubic equations,
derived from the writings of Cardan. greatest
German
Stifel
ranks as the
algebraist of the sixteenth century.
He was
educated in a monastery at Esslingen, his native place, and afterwards became a Protestant minister. Study of the
numbers in Revelation and in Daniel drew him to mathematics. He studied German and Italian significance of mystic
and in 1544 published a Latin treatise, the Arithmetical Therein he observes Integra, given to arithmetic and algebra. the advantage in letting a geometric series correspond to an wt rks,
arithmetic series, remarking that
it is
possible to elaborate a
vhole book on the wonderful properties of numbers depending n this relation. He here makes a close approach to the idea
He gives the binomial coefficients arising in and uses of the expansion (a f 6) to powers below the 18th, notations German roots. of extraction the these coefficients in of a logarithm.
are illustrated by
tiie
following :
IS ensus et 2000 aequales 680 rebus,
The
greatest French algebraist o
fea
o
sixieerffei
Fm&dsms VMa (1540~1603> He was a ntfcra of Filo and died in Paris, He was ediieatei a lawyer; Ma Manks^d 1 HI. sad Hemy IT. To spent in public service under Henry Mm mathematics was a relaxa&km. Like Naper h $&B not ?
profess to be a ma&iemalieiaa.
During the war against Spain,
he rendered service to Henry IV. by deciphering mtere^p&ed with letters written in a cipher of more than 500 characters to coori the addressed and Spanish, by rariabie signification,
A HISTORY OF MATHEMATICS
230
their governor of Netherlands.
The Spaniards
discovery of the key to magic.
Yieta
is
attributed the
said to have printed
works at his own expense, and to have distributed them among his friends as presents. His In Artem Analyticam Isagoge, Tours, 1591, is the earliest work giving a symbolical all his
Not only did he improve the algebra and trigonometry of his time, but he applied algebra to geom etry to a larger extent, and in a more systematic manner, than had been done before him. He gave also the trigonometric treatment of algebra.
j
solution of Cardan s irreducible case in cubics.
In the solution of equations Vieta persistently employed the principle of reduction and thereby introduced a uniformity
uncommon in his day. He reduces affected quad pure quadratics by making a suitable substitution for the removal of the term containing x. Similarly for cubics of treatment ratics to
and biquadratics.
Vieta arrived at a partial knowledge of
the relations existing between the coefficients and the roots of an equation. Unfortunately he rejected all except positive roots, and could not, therefore, fully perceive the relations in question.
the facts a?
(tt
is
+v+
roots u, vy w.
His nearest approach to complete recognition of contained in the statement that the equation twno has the three w)af+(uv 010 zcw)a?
+
+
Tor cubics, this statement
=
is perfect, if u, v, w, are
allowed to represent any numbers. But Vieta is in the habit of assigning to letters only positive values, so that the passage
means less than at first sight it appears to do. 1 As as 1558 Jacques Pdetier early (15171582), a French textbook writer on algebra and geometry, observed that the root of an really
equation is a divisor of the last term. Broader ^iews were held by Albert Girard (15901633), a noted Flemish mathema, tkaao,
who
in 1629 issued his Invention nouveOe en 1
H^JTKEL, p, 379.
THE RENAISSANCE
He was the
first
231
to understand the use of negative roots in the
solution of geometric problems.
He
spoke of imaginary quan
and inferred by induction that every equation has as many roots as there are units in the number expressing its He first showed hour to express the sums of their degree. titieSj
products in terms of the coefficients, The sum of the roots, giving the coefficient of the second term with the sign changed, he called the premtere faction; the sum of the products of the roots, two and two, giving the coefficient of the third term, he called deuxi&me faction, etc. In case of the equation a?
x
4
43H30,
=
1
V
he gives the roots ^ =1 ; a^=l, ;% = 2, and then states that the imaginary roots are
1+V^,
serviceable in showing the generality of the law of formation of the coefficients from the roots. 1 Similar researches on tie
theory of equations were made in England independently by Thomas Harriot (15601621). His posthumous work, the Artis Analyticce Praxis, 1631, was written long before Girard's Invent &m, though published after it. Harriot discovered the relations between the roots and the coefficients of an equation in its simplest form. This discovery was therefore made about the same time by Harriot in England, and by Vieta
and Girard on
tibe
pose equations into
ofttment. tiieir
Harriot was the
first
to decom
simple faetes, but as he failed to
recognize imaginary, or even negative roots, he failed to p^ove that every equation eotild be thus decomposed. Harriot was the earliest algeteaisfc of IkiglniML AHer grad
uating from Oxford, he remded with Sir Waiter Baleigh as mathematical tutor. 1 Raleigh sent to Virginia as surveyor in 1585 with Sir Biehard Grenville's expedition
Mm
Ms
After
Ms
return, the following year,
he published **A Brief
and True Eeport of the [Newfound Land of Virginia, 78a
*
Dictionary of NtfoiMa
m&gr^^.
A HISTORY
232 excited
muck
notice
and
Off
MATHBMA'HCS
"was translated into Latin.
Among
the mathematical instruments by which the wonder of the
Indians was aroused, Harriot mentions "a perspective glass
whereby was snowed many strange
1
sights."
time, Henry, Earl of Northumberland,
Harriot.
Admiring
his affability
and
About
this
became interested in learning, he
allowed
300 a year. In 1606 the Earl was Harriot a lifepension of committed to the Tower, but his three mathematical friends, Harriot, Walter Warner, and Thomas Hughes, "the three
magi of the Earl of Northumberland," frequently met there, and the Earl kept a handsome table for them. Harriot was in poor health,
which
explains, perhaps, his failure to complete
aad publish his discoveries. We next summarize the views regarding the negative and imaginary, held by writers of the sixteenth century and the early part of the seventeenth. Cardan's "pure minus" aad Ms views on imaginaries were in advance of his age. Until the beginning of the seventeenth century mathemati cians dealt exclusively with positive quantities.
that
"minus times minus gives
Pacioli says but plus," applies this only
to the formation of the product of (a
b)(c
negative quantities do not appear in his work. "
Cossist,"
roots,
d).
Purely
The German
Rudolf^ knows only positive numbers and positive
His + and numbers as "less than "absurd numbers," which arise when real
notwithstanding his use of the signs
.
successor, Stifel, speaks of negative
nothing," also as numbers above zero are subtracted from zero.*
Harriot
is
who
occasionally places a nega&ve term by itself on ae side of an equation. Vieta knows oaiy positive numbers,
the
first
w^
1 Harriot an asfex)nomer as well as mat^eDmtkHaji, and lie "applied e telescope to celestial purposes almost simultaneously with Galileo."
CASTOR II,
"
THE EBHAISSANCE
233
had advanced views, both on negatives and imagiBefore the seventeenth century, the majority of the great European algebraists had not quite risen to the views taught by the Hindus. Only a few can be said, like the but Girard
naries.
Hindus, to have seen negative roots j perhaps
all
Europeans,
like the Hindus, did not approve of the negative.
The
full
interpretation and construction of negative quantities and the systematic use of them begins with Rene Descartes (1596
him erroneous views respecting them appear again and again. In fact, not until the middle of the nineteenth century was the subject of negative numbers properly 1650), but after
explained in school algebras. The question naturally arises, why was the generalization of the concept of number, so as to include the negative, such a difficult step? The answer n " Negative numbers appeared absurd or "fictitious," so long as mathematicians had not hit upon a visual or graphical representation of them. The Hindus early saw in " of direction " on a line an
would seem
to be this
:
opposition
of positive and negative numbers. " debts " offered to them another
The
interpretation " ideas of " assets and
explanation of their nature. these ideas was not acquired before the time of Girard and Descartes. To Stifel is due the
In Europe
full possession of
absurd expression, negative numbers are " less than nothing." from
It took about 300 years to eliminate this senseless phrase
mathematical language. History emphasizes the importance of giving graf^kal representations of
Omit
negative
all illustration
negative numbers will
numbers in teaeMg
algebra.
by the thermometer, and be as absurd to modern stodenfe as
by
lines, or
they were to the early algebraists. In the development of symbolic algebra great services were rendered by Yieta, Epochmaking was the practice intro
duced by him of denoting general or indefinite expressions
A HISTORY OF MATHEMATICS
234
by
To be
letters of the alphabet.
part of
essential
called
by him
was written
"
algebra.
his
notation
a cubus a
cube sequalia
him
made them an The new symbolic algebra was first
in opposition to the old logistica logistica speciosa
By
numerosa.
and
sure, Stifel, Cardan,
him, but Yieta
letters before
others used
+
f b in
b cubo."
+ 3a 6 + 3a7> + 53 = (a + 6) a in b quadr. 3 + b quadr. 3
s
2
2
as
a +The vinculum was introduced by
as a sign of aggregation.
Parentheses
first
occur with
Girard.
In numerical equations the
unknown quantity was de
noted by
N
cube by 0.
9
square by Q, and
its
its
Vieta:
1(78 Q h 16 .# aequ.
Vieta:
A cubus f 5 piano 3 in A, sequari
Girard:
IDO
Descartes:
x
tf
40,
Z solido 2,
13(i)
0^
+ 12,
x
3
Illustrations
I :
 8 tf f 16 x = 40.
+ 3^ = 20. = 13a; + 12.
jp
Harriot adopted sign of equality, =, is due to Kecorde. the of in small letters of the alphabet capitals used by place 3 z 3 3 bba aaa 2 thus c ab 3 Harriot writes a Yieta,
Our
=
=
2
The
ccc.
:
symbols of inequality,
>
and <, were intro
duced by him. William Oughtred (15741660) introduced for dif for proportion, X as a symbol of multiplication, 10 A writes he In his Clavis, 1631, ference. thus, Aqqcc an^ :
~
:
;
120
A JE 7
Yincent
Z
thus, 120 AqqcEc.
Wing
Emerson
in 1651
;
The
the symbol
was used by for variation, by William
sign <x
:
for ratio
in 1768.
The
Least
Three Centuries
the building up of our modern theory of exponents and our exponential notation were taken
The
first
steps toward
by Simon Stemn (15481620) of Bruges
in Belgium.
Oresme's
previous efforts in this direction remained wholly unnoticed, 1
MATTHIESSBN, pp. 270, 37L
235
THE LAST THREE CENTTJBIES but
$tevin's
neglected at first, are a exponential notation grew in
innovations, though
His
permanent possession. connection with his notation for decimal
fractions.
Denoting
quantity by Q, he places within the circle the He a?. the of , () signify x, x?, power. Thus (T), exponent mean extends his notation to fractional exponents. 0, , (),
the
unknown
x^9 x*,
M
x$.
He
writes 3xyz2 thus 3(T)
means multiplication;
The
Q
M
second;
sec,
sec ter,
M ter@, where
third
unknown
x was adopted by Girard. Stevin's quantity. great independence of mind is exhibited in his condemnation of such terms as "sursolid," or numbers that are "absurd," for
"irrational," "irregular," "surd."
He
shows that
all
num
bers are equally proper expressions of some length, or some power of the same root. He also rejects all compound ex
such as "squaresquared," "cubesquared," and " that they be named by their exponents the fourth," suggests failed to unknown for the ," fifth" powers. Stevin's symbol pressions,
be adopted, but the principle of his exponential notation has The modern formalism took its shape with Des survived.
In his Geometry, 1637, he uses the last letters of the z to designate alphabet, x in the first place, then the letters y,
cartes.
unknown Quantities while the first letters of the alphabet are made to stand for known quantities. Our exponential nota ;
*
4
tion,
a
,
is found in Descartes
;
however, he does not use gen
n
nor negative and fractional ones. In In case this last respect he did not rise to the ideas of Stevin. of radicals he does not indicate the root by indices, but in eral exponents, like
case
of
cube
root,
VaTI^A/H
a
,
for
instance,
uses the
letter
C, thus,
1
early notations for evolution, two have come down to our time, the German radical sign and Stevin's fractional
Of the
i
CJOTOR,
II, 794.
A HISTORY OF MATHEMATICS
236
exponents. Modern pupils have to learn the algorithm for both notations ; they must learn the meaning of fya*, and also of its equal
(A
It is a great pity that this should be
The
so.
operations with fractional exponents are not always found easy,
and the
" rules for radicals are always pronounced hard."
By
learning both, the progress of the pupil is unnecessarily re Of the two, the exponential notation is immeasurably tarded.
Radicals appear only in evolution. 1 Exponents, on the other hand, apply to both involution and evolution with superior.
;
them
all operations
parative ease.
we
and
simplifications are effected with
In case of
what a gain
radicals,
could burst the chains which
tie
would
it
us to the past
com be, if
!
Descartes enriched the theory of equations with a theorem which goes by the name of " rule of signs." By it are deter
mined the number of positive and negative roots of an equation the equation may have as many roots as there are variations in sign, and as many roots as there are permanences in sign*:
+
Deseartes was accused by Wallis of availing himself, without
acknowledgment, of Harriot's theory of equations, particularly Ms mode of generating equations; but there seems to be no
good ground for the charge. Wallis claimed, moreover, that Descartes failed to notice that the rule breaks down in case of imaginary
roots,
but Descartes does not say that the equation
always has, but that
it
may
have, so
many
does not consider the
that Descartes
It is true
roots.
case
of
imaginaries
but further on, in his Geometry, he gives ample evi of his ability to handle the case of imaginaries. dence directly
;
*'
V
In connection with the imaginary, 1, the radical notation is objedaonahle, because it leads students and even authors' to "remark that the rules of operation for real quantities do not always hold for imagi1
Baiies, since
V
1

V
1
V+
does not equal 1. That the difficulty is from the fact that it disappears when
Mtereiy one of notation is evident
we
le8%aa&e the imaginary
definition,
equak
 1.
*Hift
by
**
Then
i
i
= &,
which, hy
237
THE LAST THREE CENTURIES
was an English mathematician of educated for the Church, at Cam great originality. in 1649 was appointed bridge, and took Holy Orders, but He advanced Savilian professor of geometry at Oxford. John, Wallis (16161703)
He was
" law of beyond Kepler by making more extended use of the continuity/ applying it to algebra, while Desargues applied it to geometry. By this law Wallis was led to regard the 7
denominators of fractions as powers with negative exponents. 2 For the descending geometrical progression a? x\ as if con ,
1 tinued, gives or ,
etc.
a?
2
The exponents
metical progression
,
etc.
;
which
is
,
the same thing as ,
^,
of his geometric series are in the arith 2. He also used fractional 1,
2, 1, 0,
but had exponents, which had been invented long before, The symbol oo for infinity failed to be generally introduced. is
due to him.
In 1685 Wallis published an Algebra which
has long been a standard work of reference. It treats of the The history, theory, and practice of arithmetic and algebra.
and worthless, but in other respects the work is a masterpiece, and wonderfully rich in material. The study of some results obtained by Wallis on the quad historical part is unreliable
Newton to the discovery of the Binomial made about 1665, and explained in a letter written Theorem, 1 Newton's reason to Newton Oldenburg on June 13, 1676. by n n be positive whether of the (a + 6) development ing gives n is a positive when fractional. or or negative, integral Except
rature of curves led
,
He gave no regular integer, the resulting series is infinite. actual multiplication. proof of his theorem, but verified it by The
case of positive integral exponents was proved
by James
Bernoulli * (16541705), by the doctrine of combinations.
A
1 How, tke Binomial Theorem was deduced as a corollary of WaJHs*a results is explained in Cajori's History of Mathematics, pp. 195, 196, Cwjectandi, 1713, p. 89. 1
A HISTORY OF MATHEMATICS
238
proof for negative and fractional exponents was given by Leonhard Euler (17071783), It is faulty, for the reason that
he
fails to consider the convergence of the series nevertheless has been reproduced in elementary textbooks even of recent 1 A rigorous general proof of the Binomial Theorem, years. ;
it
embracing the case of incommensurable and imaginary powers, was given by Niels Henrik Abel. 2 It thus appears that for over a century and a half this fundamental theorem went without 3
adequate proof. Sir Isaac Newton (16421727) matical
mind
of all times.
is
Some
probably the greatest mathe idea of his strong intuitive
be obtained from the fact that as a youth he regarded the theorems of ancient geometry as selfevident truths, and that, without any preliminary study, he made
powers
may
himself master of Descartes' Geometry. He afterwards re this of garded neglect elementary geometry as a mistake, and
once expressed his regret that "he had applied himself to the works of Descartes and other algebraic writers before he had considered the Elements of Euclid with the attention that so excellent a writer deserves."
During the
first
nine years of
For the history of Infinite Series see REIFF, Geschichte der Unendlichen Iteihen, Tubingen, 1889; CANTOR, HI., 5394; CAJORI, pp. 334339 ; Teach, and Hist, of Math, in the U.S., pp. 361376. 1
2
See CRELLE,
I.,
1827, or CEuvres completes, de N. H.
ABEL, Chris
tiania, 1839, 1., 66 et seq. 5 It
should be mentioned that the beginnings of the Binomial Theorem
for positive integral exponents are found very early. The Hindus and s Arabs used the expansions of (a f &) 2 and (a 6) hi the extraction of 4 square and cube root. Vieta knew the expansion of (a f 5) . But these
+
were obtained by actual multiplication, not by any law of expansion. Stifel gave the coefficients for the first 18 powers ; Pascal did similarly in his "arithmetical triangle" (see CANTOR, II., 685, 686). Pacioli, Stevin, Briggs, and others also possessed something, from which one would think the Binomial Theorem could have been derived with a little " if we did not know that such attention, simple relations were difficult to discover " (De Morgan).
THE LAST THREE CENTURIES
239
Cambridge he delivered lectures on alge Over bra. thirty years after, they were published, in 1707, by Mr. Whiston under the title, Aritlimetica Universalis. They his professorship at
contain
new and important
results
on the theory of equations. is well known.
His theorem on the sums of powers of roots
We give a specimen of his notation a + 2 aac aab' 3 :
8
abc
+ bbc.
Elsewhere in his works he introduced the system of
The
literal
Universalis contains also a large number of problems. Here is one (No. 50) " stone falling down into a well, from the sound of the stone striking the
indices.
AritJimetica
:
A
He leaves his bottom, to determine the depth of the well." problems with the remark which shows that methods of teach ing secured some degree of attention at his hands: "Hitherto I have been solving several problems. For in learning the sciences examples are of
more use than precepts." l
1 Newton's body was interred in Westminster Abbey, where in 1731 a fine monument was erected to his memory. In cyclopaedias, the statement is frequently made that the Binomial Formula was engraved upon Newton's tomb, but this is very probably not correct, for the
We
have the testimony of Dr. Bradley, the following reasons: (1) Bean of Westminster, and of mathematical acquaintances who have visited the
Abbey and mounted
monument, that the theorem can Yet all Latin inscriptions None of the biographers of Newton
the
not be seen on the tomb at the present time. are
still
distinctly readable.
(2) of the old guidebooks to Westminster Abbey mention the Binomial Formula in their (often very full) descriptions of Newton's
and none
However, some of them say, that on a small scroll held by two winged youths in front of the half recumbent figure of Newton, there are some mathematical figures. See Neale's Guide. Brewster, in his Life of Sir Isaac Newton, 1831, says that a "converging series" is there, but this does not show now. Brewster would surely have said "Binomial Theorem " instead of " converging series " had the theorem been there. The Binomial Formula, moreover, is not always convergent. (3) It is important to remember that whatever was engraved on the scroll could not be seen and read by any one, unless he stood on a chair or tomb.
A HISTORY OF MATHEMATICS
240
The
principal
investigators on the
solution of numerical
Kewton, Lagrange, Joseph Fourier, and Before Vieta, Cardan applied the Hindu rule Homer. of "false position" to cubics, but his method was crude. Vieta gave a process which is laborious, but is the forerunner equations are Yieta,
1 The later of the easy methods of Newton and Homer. as so to afford in are the the of work, arrangement changes
Homer's Wittiam George Horner process is the one usually taught. (17861837), of Bath, the son of a Wesleyan minister, was His method educated at Kingswood School, near Bristol. was read before the Royal Society, July 1, 1819. 2 Almost precisely the same method was given by the Italian Paolo Buffini in 1804. Neither Ruffini nor Horner knew that their method had been given by the Chinese as early as the thir teenth century. Be Morgan admired Hornets method and per fected it in details still further. It was his conviction that it should be included in the arithmetics; he taught it to his classes, and derided the examiners at Cambridge who ignored facility
and security in the evolution of the
the method. 3
De Morgan
root.
encouraged students to carry out
used a stepladder. Whatever was written on the scroll was, therefore, not noticed "by transient visitors. The persons most likely to examine everything carefully would be the writers of the guidebooks and the the very ones who are silent on the Binomial Theorem. On biographers the other band, such a writer as E. Stone, the compiler of the New Math ematical Dictionary, London, 1743, that the theorem is "put upon the
is more likely to base his statement monument," upon hearsay. (4) We
have the positive testimony of a most accurate writer, Augustus Be Morgan, that the theorem is not inscribed. Unfortunately we have no where seen his reasons for the statement See his article " Newton " in the Penny or the English Cyclopaedia. See also our article in the Bull, of ike Am. Math. Soc., L, 1894, pp. 5254. 1 Oolo. College Publication* General Series 51 and "52, 1910. * DicUona^ of National Biograpit. * See DIB MoBAix, Bwfyet of Paradoxes, 2d Ed., 1915, VoL IL, pp. 188,
THE LAST THREE CENTUBIES
241
long arithmetical computations for the sake of acquiring skill 2x 5 to and rapidity. Thus, one of his pupils solved x3 " another tried 150 but broke down 103 decimal
=
places,
places,
at the 76th,
which was wrong." l
While, in our opinion,
De
Morgan greatly overestimated the value of Homer's method to the ordinary boy, and, perhaps, overdid in matters of it
calculation,
have gone
certainly true
is
that in America teachers
to the other extreme, neglecting the art of rapid
computation, so
that our
school
children
have been con
spicuously wanting in the power of rapid and accurate fig 2
uring.
In
1
TT.
The
we
consider the approximations to the early European computers followed the
this connection
value of
GRAVES, Life of Sir
Wm. Eowan
Hamilton^ HL,
p. 275.
by Mr. E. M. Langley in the Eighteenth Gen " On eral Report of the A. L G. T., 1892, p. 40, from De Morgan's article Arithmetical Computation" in the Companion to the British Almanac for 1844, is interesting: "The growth of the power of computation on the Continent, though considerable, did not keep pace with that of the same in England. We might give many instances of the truth of our In 1696 De Lagny, a wellknown writer on algebra, and a assertion. member of the Academy of Sciences, said that the most skilful computer could not, in less than a month, find within a unit the cube root of 696536483318640035073641037. We would have given something to have been present if De Lagny had ever made the assertion to his contempo 2
The
following, quoted
Abraham Sharp. In the present day, however, both in our uni homo and everywhere abroad, no disposition to encourage computation exists among those who attend to the higher branches of
rary,
versities at
mathematics, and the elementary works are very deficient in numerical examples." De Lagny s example was brought to De Morgan in a class, and he found the root to five decimals in less than twenty minutes. Mr. J
Langley exhibits De Morgan's computation on p. 41 of the article cited. Mr. Langley and Mr. R. B. Hayward advocate Homer's method as a " the desirable substitute for clumsy rules for evolution which the young student still usually encounters in the textbooks." See HATWABB'S the A. L
article in article,
A HISTORY OF MATHEMATICS
242
geometrical method of Archimedes by inscribed or circum scribed polygons. Thus Vieta, about 1580, computed to ten places, places, latter
Adrianus Romanus (15611615), of Louvain, to 15 Ludolph van Ceulen (15401610) to 35 places. The spent years in this computation, and his performance
was considered so extraordinary that the numbers were cut on his tombstone in St. Peter's churchyard at Leyden. The tombstone
but a description of
it is extant. After him, often called " Ludolph's number." In the seventeenth century it was perceived that the computations
is lost,
the value of
TT
is
could be greatly simplified by the use of infinite
a
series,
gested by
viz.
1
tan x
=x
~f
3
James Gregory
7
5
H ,
series.
was
first
Such sug
Perhaps the easiest are the Machin's formula is,
in 1671.
formulae used by Machin and Base.
The Englishman, Abraham Sharp, a
skilful
mechanic and
computer, for a time assistant to the astronomer Flamsteed, took the arc in Gregory's formula equal to 30, and calculated
x to 72 places in 1705 next year Machin, professor of astron omy in London, gave TT to 100 places; the Frenchman, Be ;
Lagny, about 1719, gave 127 places in 1793, 140 places
;
places (152 correct);
the German, Georg Yesa, the English, Rutherford, in 1841, 208 the German, Zacharias Dase, in 1844, ;
205 places ; the German, Th. Clausen, in 1847, 250 places ; the English, Rutherford, in 1853, 440 places William Shanks, in ;,
1 1873, 707 places.
It
may
be remarked that these long com
putations are of no theoretical or practical value.
Infinitely
more interesting and useful are Lambert's proof of 1761 1
W. W.
R. BALL, Math. RevreaMons and Problems, pp. 171l?a
Ball gives MbliograpMcal references.
THE LAST THREE CESTTUBIES
248
not rational/ and Lindemann's proof that TT is not algebraical, i.e. cannot be the root of an algebraic equation. that
IT
is
by which TT may be computed were given also by Hutton and Euler. Leonhard Euler (17071783), of Basel, Infinite series
contributed vastly to the progress of higher mathematics, but his influence reached down to elementary subjects. He treated
trigonometry as a branch of analysis, introduced (simultane ously with Thomas Simpson in England) the now current abbreviations for trigonometric functions, and simplified for mulae by the simple expedient of designating the angles of
a triangle by A, B,
0,
his old age, after he
and the opposite sides by a, 5, had become blind, he dictated
c.
In
to his
servant his Anleitung zur Algebra, 1770, which, though purely elementary, is meritorious as one of the earliest attempts to put the fundamental processes upon a sound basis.
An Introduction
Elements of Algebra, selected from the Algebra of Euler, was brought out in 1818 by John Farrar of Harvard College. to the
.
.
.
A
question that became prominent toward the close of the eighteenth century was the graphical representation and inter 1. As with negative numbers, pretation of the imaginary,
V
no decided progress was made until a was presented to the eye. In the time of Newton,
so with imaginaries,
picture of
it
Descartes, and Euler, the imaginary was still an algebraic The first thoroughly successful graphic representations
fiction.
were described by the Norwegian surveyor, Caspar Wessel, in an article of 1797, published in 1799 in the Memoirs of the Danish E.
Academy of Sciences, and by Jean Robert Argand (1768?), who in 1806 published a remarkable Essai* But
of Geneva,
1 See the proof in Note IV. of Legendre's QeonUtrie, where tended to w2
it is
ex
.
2 Consult Their Geometrical Interpretation, Imaginary Quantities. Translated from the Erench of M. ABQJLNIX by A. S. HARDY,
York, 1881.
A HISTORY OF MATHEMATICS
244 all
these writings were
little
noticed,
and
it
remained for the
great Carl FriedricJi Gauss (17771855), of Gottingen, to break He introduced the last opposition to the imaginary. it as an independent unit coordinate to 1, and a f ib as a
down
"complex number." Notwithstanding the acceptance of imagi"numbers" by all great investigators of the nine teenth century, there are still textbooks which represent the naries as
obsolete view that
V
1 is not a
number
or is not a quantity.
Clear ideas on the fundamental principles of algebra were As late as the not secured before the nineteenth century. latter part of the eighteenth century
we
find at Cambridge,
1 England, opposition to the use of the negative.
The view was held that there exists no distinction between arithmetic and algebra. In fact, such writers as Maclaurin, Saunderson,
Thomas Simpson, Hutton,
Bonnycastle, Bridge, began their with arithmetical algebra, but gradually and disEarly American guisedly introduced negative quantities. writers imitated the English. But in the nineteenth century
treatises
the
principles of algebra
first
gated by George Peacock,
2
came
to be carefully investi
D. F. Gregory, 3
De Morgan. 4
Of
we may mention Augustin Louis Cauchy Martin Ohm, 6 and especially Hermann Hankel. 7
continental writers 5
(17S91S57),
See C.WORDSWORTH, Scholce Academics : Some Account of the Studies Universities in the Eighteenth Century, 1877, p. 68 ; Teach, and Hist, of Math, in the U. S. pp. 385387. " 2 See his Report on Recent Progress Algebra, 1830 and 1842, and his in Analysis," printed in the Reports of the British Association, 1833. 8 ** On the Real Nature of Symbolical Algebra," Trans. Hoy. Soc, 1
at English
9
Edinburgh, Vol. XIV., 1840, 4 u On the Foundation of 1S41, 1842 &
;
p. 280.
Algebra," Cambridge Phil. Trans., VIL,
VIIL, 1844, 1847.
Analyse Algebrique, 1821, p. 173 et seq. Versitchs eines vollkommen consequenten Systems der Mathematik, Ed. 1828. 1822, ' &&e This work is very rich in Comjflexen ZaMen, Leipzig, 1807. listeieal notes. Most of the bibliographical references on this subject given here axe taken from that work. 6
M
245
EDITIONS OF EUCLID
A
flood of additional light has been
thrown on
this subject
Rowan Hamilton, Hermann Grassmann, and Benjamin Peirce, who conceived by the epochmaking researches of William
new
algebras with laws differing from the laws of ordinary 1
algebra.
GEOMETRY
AJSTD
TEIGOFOMETRl
Editions of Eudid.
With the
Early Researches
and beginning of Great progress was
close of the fifteenth century
the sixteenth
we
enter
upon a new
era.
in arithmetic, algebra, and trigonometry, but less prom inent were the advances in geometry. Through the study of
made
Greek manuscripts which, after the fall of Constantinople in 1453, came into possession of Western Europe, improved trans At the beginning of this lations of Euclid were secured. period, printing
The
ful.
first
Venice, 1482.
Campanus.
books became cheap and plenti ; of Euclid was published in edition printed the translation from the Arabic by This was
was invented
Other editions of this appeared, at TJlm in 1486, The first Latin edition, translated from
at Basel in 1491.
the original Greek, by Bartholomcei^s Zambertus, appeared at In it the translation of Campanus is severely
Venice in 1505.
This led Pacioli, in 1509, to bring out an edition, the tacit aim of which seems to have been to exonerate Cam criticised.
2 Another Euclid edition appeared in Paris, 1516. The panus. first edition of Euclid printed in Greek was brought out in
Basel in 1533, edited by Simon Grrynceus. For 170 years this was the only Greek text In 1703 Dawd Gregory brought out at Oxford all the extant works of Euclid in the original 1
For an
GIBBS, 2
excellent historical sketch
on Multiple Algebra, see
J.
W.
m Proceed. 4m. Ass. for the Adv. of Science, VoL XXXV., 1886.
CXNTOR, IL,
p. 312.
A HISTORY OF MATHEMATICS
246
As a complete edition of Euclid, this stood alone until 1883, when Heiberg and Menge began the publication, in Greek The first and Latin, of their edition of Euclid's works. in of 1570 translation the made from Elements was English 1 An the Greek by "EL Billingsley, Citizen of London." English edition of the Elements and the Data was published in 1758 by Robert Simson (16871768), professor of mathe matics at the University of Glasgow. His text was until recently the foundation of nearly all school editions. It dif fers considerably from the original. Simson corrected a numoer of errors in the Greek copies. All these errors he assumed due to unskilful editors, none to Euclid himself. close
A
to be
English translation of the Greek text was made by James WiUiamson. The first volume appeared at Oxford in 1781 ;
the second volume in 1788. usually contain the
first six
School editions of the Elements books, together with the eleventh
and twelfth. Returning to the time of the Eenaissance, we mention a few iln the General Dictionary "by BAYLB, London, 1735, it says that Billingsley "made great progress in mathematics, by the assistance of his friend, Mr. Whitehead, who, being left destitute upon the dissolution of the monasteries in the reign of Henry VIII., was received by Billingsley into his family,
and maintained by him
in his old age in his house at
London."
Billingsley was rich and was Lord Mayor of London in 1591. Like other scholars of his day, he confounded our Euclid with Euclid of
Megara. The preface to the English edition was written by John Dee, a famous astrologer and mathematician. An interesting account of I>ee is given in the Dictionary of National Biography. De Morgan thought that Dee had made the entire translation, but this is denied in the article " 'BUHngsley of this dictionary. At one time it was believed that Bil lingsley translated from an ArabicLatin version, but G. B. Halsted suc ceeded in proving from a folio once the property of Billingsley [now in the library of Princeton College, and containing the Greek edition of 1533, together with some other editions] that Billingsley translated from t&a Greek, not the Latin. See "Note on the First English Euclid" in t&e
Am.
Jowr* of Jfo&em.,
ToL IL, 187&
EABLY RESEARCHES of the
247
more interesting problems then discussed in geometry.
of new geometrical methods of investigation we find as yet no trace. In his book, De triangulis, 1533, the German astronomer, Regiomontanus, gives the theorem (already known to Proclus) that the three perpendiculars from the ver
Of a development
meet in a point and shows how to find from the three sides the radius of the circumscribed circle. He gives the first new maximum problem considered since the time of tices of a triangle
;
Apollonius and Zenodorus, viz., to find the point on the floor (or rather the locus of that point) from which a vertical 10 foot rod,
whose lower end
4 feet above the
is
floor,
seems
subtends the largest angle). 1 3sTew is the following theorem, which brings out in bold relief a fundamental differ ence between the geometry on a plane and the geometry on a
largest
(i.e.
Prom
the three angles of a spherical triangle may be the three Eegiomontanus dis sides, and vice versa. computed cussed also starpolygons. He was probably familiar with the
sphere
:
writings on this subject of
Campanus and Bradwardine. Eegi
omontanus, and especially the Frenchman, Charles de Bouvelles, or Carolus Bbvillus (14701533), laid the foundation for the 2 theory of regular starpolygons. The construction of regular inscribed polygons received the
S.
1
CANTOR,
2
A
II., 283.
and GUNTHER, Vermischte Untersuchungen, pp. detailed history of starpolygons
commanded
the attention of geometers
starpolysedra is given by 192. Starpolygons have
down
even to recent times.
the more prominent are Petrus Ramus, Athanasius Kircher (16021680), Albert Girard, John Broscius (a Pole), John Kepler, A, L. F. Meister (17241788), C. F. Gauss, A. F. Mo'bius, L. Poiusot (1777
Among
Mobius gives the following definition for the area 1859). C. C. Krause. of a polygon, useful in case the sides cross each other: Given an arbi
P
. MN; assume any point in the trarily formed plane polygon AB . plane and connect it with, the vertices by straight lines; then the sum is independent of the position oj PAB + + . . + .
PBC
.
PMN+ PNA
P and represents the area of the polygon.
Here
PAB =
PBA,
A HISTORY OF MATHEMATICS
248
attention of the great painter and architect, Leonardo da Vinci (14521519). Of his methods some are mere approximations,
of no theoretical interest, though not without practical value.
His inscription of a regular heptagon (of approximation) he considered to be accurate tions were given
an
Similar construc
painter, Albreclit Diirer,
who always
and correctly which of the constructions are approximations. 1 Both
(14711528). states
by the great German
He
course, merely !
is
the
first
clearly
Leonardo da Vinci and Diirer in some cases perform a construc tion by using one single opening of the compasses. Pappus
Abul "Wafa did this repeat method becomes famous. It was used by Tartaglia in 67 different constructions it was employed also 2 by Ms pupil Giovanni Battista Benedetti (15301590), It will be remembered that Greek geometers demanded that all geometric constructions be effected by a ruler and compasses only other methods, which have been proposed from time to once set himself this limitation edly; but
now
;
this
;
;
time, axe to construct 8 only, or
by
by the compasses only or by the ruler and other additional instruments.
ruler, compasses,
Constructions of the last class were given by the Greeks, but were considered by them mechanical, not geometric. A peculiar feature in the theory of all these methods is that elementary 1
2
CANTOR, IL, 462. For further details, consult CANTOR
S.
GONTHEE, Nachtrage, pretty method is reached
p.
117, etc.
II.,
The
296300, 460, 527, 529
fullest
j
development of this
in STEINER, Die Gfeometrisclien Constructionen, ausgefuhrt mittels der geraden Linie and eines festen Kreises, Berlin, 1833 ; and in PONCELET, Traite des proprietes projectiles, Paris, 1822,
p, 187, etc. s
Problems to be solved by aid of the ruler only are given by LAMBERT ; by SERVOIS, Solutions peu connues de differens proolemes de Geometrie pratique, 1805 ; BRIANCHON, M&^ire swr ropp&eo&ora de la theorie des transversales. See also CHASLES, pw 210 j CfcEKOHA, Elements of Projectwe Geometry, Transl. by LEHDESin his Freie Perspective, Zurich, 1774
F,
Oodtoi, 1885, pp. xiL, 9698.
EABLY RESEARCHES geometry
is
249
unable to answer the general question, What con by either one of these methods ?
structions can be carried out
For an answer we must resort
to algebraic analysis.
1
A construction by other instruments than merely the ruler and compasses appears in the quadrature of the circle by Leo nardo da Vinci. He takes a cylinder whose height equals hali its
radius
;
its
trace
on a plane, resulting from one revolution, circle. Nothing it must not be
a rectangle whose area is equal to that of the could be simpler than this quadrature; only
is
claimed that this solves the problem as the Greeks understood The ancients did not admit the use of a solid cylinder as an it.
instrument of construction, and for good reasons while with a ruler we can easily draw a line of any length, and, with an or :
dinary pair of compasses, any circle needed in a drawing, we can with a given cylinder effect not a single construction of No draughtsman ever thinks of using a practical value. 2
cylinder.
To Albrecht Diirer belongs the honour of having shown how the regular and the semiregular solids can be constructed out of paper by marking off the bounding polygons, all in one and then folding along the connected edges. 8 Polyedra were a favourite study with John Kepler.
In
1596, at the beginning of his extraordinary scientific career,
he
piece,
pseudodiscovery which brought him much fame. He placed the icosaedron, dodecaedron, octaedron, tetraedron, and cube, one within the other, at such distances that each polye
made a
dron was inscribed in the same sphere, about which the next was circumscribed. On imagining the sun placed in
outer one
the centre and the planets moving along great circles on the taking the radius of the sphere between the icosaespheres ,
p. 2.
For a good article on the " Squaring SCHDBBKT in Monist^ Jan., 1891. 2
of the Circle," see 8
CANTOR, IL, 466.
A HISTORY OF MATHEMATICS
250
dron and dodecaedron equal to the radius of tlie earth's orbit he found the distances between these planets to agree roughly This reminds us of Pythag But maturer reflection and intercourse orean mysticism. with Tycho Brahe and Galileo led him to investigations and " results more worthy of his genius Kepler's laws*" Kepler with astronomical observations.
1 An innova greatly advanced the theory of starpolyedra. tion in the mode of geometrical proofs, which has since
been widely used by European and American writers of elementary books, was introduced by the Frenchman, Ifranciscus Yieta. He considered the circle to be a polygon with
an
infinite
Kepler.
number
of sides. 2
In recent times
The same view was taken by
this geometrical fiction has
been
generally abandoned in elementary works; a circle is not a polygon, but the limit of a polygon. To advanced mathema ticians Yieta's idea is of great service in simplifying proofs,
and by them
The
it
may
be safely used.
revival of trigonometry in
Germany
is chiefly
due to
John Mutter, more generally called Regiomontanus (14361476). At Yienna he studied under the celebrated Georg Purbach, who began a translation, from the Greek, of the Almagest,
which was completed by Eegiomontanus.
The
latter also
translated from the Greek works of Apollonius, Archimedes, and Heron. Instead of dividing the radius into 3438 parts,
Hindu fashion, Eegiomontanus divided it into 600,000 equal parts, and then constructed a more accurate table of sines. Later he divided the radius into 10,000,000 parts. The tangent had been known in Europe before this to the
in
Englishman, Bradwardine, but Regiomontanus went a step He published a further, and calculated a table of tangents. 1
For drawings of Kepler's starpolyedra and a detailed history of the see S. GUJTTHBB, Verm. Unt&rsuckungen pp. 3692.
sotofect,
y
II., 586.
EARLY RESEARCHES
251
on trigonometry, containing solutions of plane and spherical triangles. The form which he gave to trigonometry treatise
has been retained, in
The task
its
main
features, to the present day.
of computing accurate tables
was continued by
the successors of Regiomontanus. More refined astronomical instruments furnished observations of greater precision and necessitated the computation
trigonometric functions.
of
more extended
Of the several tables
tables
of
calculated,
that of Qeorg Joachim of Feldkirch in Tyrol, generally called In one of his sinetables, Wiasticus, deserves special mention.
he took the radius
= 1,000,000,000,000,000 and proceeded from
He
began also the construction of tables of For twelve years he had in continual
10" to 10".
tangents and secants.
employment
The work was completed by A republication was made These tables are a gigantic monument
several calculators.
his pupil, Valentin Otho, in 1596.
by Pitiscus in of
German
1613.
diligence
and perseverance.
But Hhaeticus was
not a mere computer. Up to his time the trigonometric functions had been considered always with relation to the arc; he was the first to construct the right triangle and to
make them depend
It was from the directly upon its angles. he took his idea of calculating the hypot His was the first European table of enuse, i.e. the secant. Good work in trigonometry was done also by Vieta, secants.
right triangle that
Adrianus Romanus, Nathaniel Torporley, John Napier, Willebrord Snellius, Pothenot, and others. An important geodetic given a terrestrial triangle and the angles subtended by the sides of the triangle at a point in the same plane, to was solved find the distances of the point from the vertices
problem
and again by Pothenot in 1730. and it secured the name was forgotten investigation The word " radian" is due to "Pothenot's problem."
by Snellius in a work Snellius of
7
of 1617,
s
James Thomson,
1871,
A
252
HISTORY OF MATHEMATICS
The Beginning of Modern Synthetic Geometry
About the beginning of the seventeenth century the
first
decided advance, since the time of the ancient Greeks, was
Two lines of progress are noticeable in Geometry. marked out by the genius of Descartes, the path, analytic (1) the inventor of Analytical Geometry ; (2) the synthetic path, made
:
with the new principle of perspective and the theory of trans
The
versals.
early investigators in
try are Desargues, Pascal, and
De
modern synthetic geome
Lahire.
Girard Desargues (15931662), of Lyons, was an architect Under Cardinal Eichelieu he served in the
and engineer. siege of
La Boehelle,
in 1628.
Soon
he
after,
where he made his researches in geometry.
retired to Paris,
Esteemed by the
ablest of his contemporaries, bitterly attacked
to appreciate his genius, his ten,
and
his
name
by others unable works were neglected and forgot
fell into oblivion until,
the nineteenth century,
it
in the early part of
was rescued by Brianchon and
Desargues, like Kepler and others, introduced the doctrine of infinity into geometry. 1 He states that the straight
Poncelet.
line
may be
regarded as a circle whose centre
is
at infinity
;
hence, the two extremities of a straight line may be considered as meeting at infinity; parallels differ from other pairs of lines only in
having their points of intersection at infinity, the gives theory of involution of six points, but his definition of " involution " is not quite the same as the modern 2 definition,' first found in Eermat, but really introduced into
He
3 geometry by Chasles.
On
a line take the point
{soucke), take also the three pairs of points
D
Gff 1
and F\ then, says Desargues,
CHARLES TATLOK, Introduction
/ Oomcs^ Cambridge, *
1881, p.
M.
if
A
as origin
and H,
Q
and
AB AH = AC AG
to the Antient *
and Modern Geometry
CAITFOR,
ConsoM CHAOLES, Note X, j MA.BIB, in.,
B
214.
II., 677, 678.
SYNTHETIC GEOMETRY
= AD on the
AF, the
six points are in " involution."
origin, then
from the
origin.
258 If a point falls
partner must be at an infinite distance from any point P lines be drawn through
its
If
MN
in six other the six points, these lines cut any transversal points, which are also in involution; that is, involution is a
Desargues also gives the theory of polar Theorem" in elementary
protective relation. lines.
works
What is
is
called "Desargues'
as follows
:
If the vertices of
either in space or in a plane, lie
two
on three
triangles, situated lines
meeting in a
H.
then their sides meet in three points lying on a line, and conversely. This theorem has been used since by Brianchon,
point,
Sturm, Gergonne, and others.
Poncelet
made
it
the basis of
his beautiful theory of homological figures. Although the papers of Desargues fell into neglect, his ideas
were preserved by his
The
latter, in 1679,
disciples, Pascal
and Philippe de Lahire.
made a complete copy of Desargues'
princi
pal research, published in 1639. Blaise Pascal (16231662) was one of the very few contemporaries who appreciated the worth of Desargues. He says in his Essais pour les eoniques, " I wish to that I owe the little that I have dis
acknowledge covered on this subject to his writings/' Pascal's genius for geometry showed itself when he was but twelve years old. His father wanted Krr\ to learn Latin and Greek before enter
All mathematical books were hidden ing^on mathematics* 0nt of sight. In answer to a question, the boy was told by
A HISTORY OF MATHEMATICS
254 his
"was
father that mathematics
the method of making
figures with exactness, and of finding out what proportions they relatively had to one another." He was at the same time
forbidden to talk any more about it. But his genius could not be thus confined meditating on the above definition, he drew figures with a piece of charcoal upon the tiles of the ;
pavement. He gave names of his own to these figures, then formed axioms, and, in short, came to make perfect
In
demonstrations.
this
way he
arrived,
unaided, at
the
theorem that the anglesum in a triangle is two right angles. His father caught him in the act of studying this theorem, and
was so astonished at the sublimity and force of his genius as weep for joy. The father now gave him Euclid's Elements, which he mastered easily. Such is the story of Pascal's early
to
1 While this narra boyhood, as narrated by his devoted sister. tive must be taken cum grano salis (for it is highly absurd to suppose that young Pascal or any one else could rediscover
geometry as far as Euclid L, ajid hitting
tration enabled
it
same treatment
upon the same sequence of propositions as found
in the Elements), conies
32, following the
it is
him
true that Pascal's extraordinary pene
at the age of sixteen to write a treatise
which passed for such a surprising
was said nothing equal to
it
since the time of Archimedes.
in power
on
effort of genius that
had been produced
Descartes refused to believe
was written by one so young as Pascal. This treatise is now lost. Leibniz saw it in Paris, recommended its publication, and reported on a portion of its
that
it
was never published, and
contents. 2
However, Pascal published in 1640, when he was sixteen years old, a small geometric treatise of six octavo 1
The Life of Mr. Paschal, by MJLDXM PERIER. Translated into Eng by W. A., London, 1744. 2 See letter written by Leibniz to Pascal's nephew, August 30, 1676,
lish
which
is
given in Oeuvres completes de Elaise Pascal, Paris, 1866, Vol.
SYNTHETIC GEOMETRY
255
Constant pages, bearing the title, Essais pour les comques. health. Pascal's tender at a age greatly impaired application
During
his adult life
he gave only a small part of his time to
the study of mathematics.
two
Pascal's
treatises just
noted contained the celebrated
proposition on the mystic hexagon, known as "Pascal's Theorem," viz. that the opposite sides of a hexagon inscribed in a conic intersect in three points which are collinear. In
our elementary textbooks on modern geometry this beautiful theorem is given in connection with a very special type of a As, in one sense, any two straight conic, namely, the circle. lines
be looked upon as a special case of a conic, the
may
theorem applies to hexagons whose first, third, and fifth ver tices are on one line, and whose second, fourth, and sixth ver tices are
on 'the
It is
other.
special case of "Pascal's
interesting to note that this
Theorem"
occurs already in Pappus Pascal said that from his theorem
(Book YIL, Prop. 139). he deduced over 400 corollaries, embracing the conies of Apollonius and many other results. Pascal gave the theorem
on the fruitful
found in Pappus. 1 This wonderfully be stated as follows: Pour lines in a
cross ratio, first
theorem
may
plane, passing through one common point, cut off four seg ments on a transversal which have a fixed, constant ratio, in
whatever manner the transversal
may
be drawn j that
is,
if
the transversal cuts the rays in the points A, B, C, D, then the ratio
BD,
:
is
,
formed by the four segments AC, AD, BC,
the same for all transversals.
The
researches
of
Desargues and Pascal uncovered several of the rich treasures The Es&ais pour les coniques is given in Vol. ITL, pp. pp. 46&4$8. 182186, of the Oeuvres completes, also in Oeuvres de Pascal (The Hague, 1779) and by H. WEISSENBORN in the preface to his book, Die Projection in der Ebene, Berlin, 1862. 1
Bmk TIL,
129,
Consult CHASLES, pp. 81, 32.
A HISTORY OF MATHEMATICS
256
modern synthetic geometry; but owing
of
interest taken in the analytical
to the absorbing
geometry of Descartes, and,
in the differential calculus, the subject was almost ei> tirely neglected until the close of the eighteenth century. Synthetic geometry was advanced in England by the re later,
searches of Sir Isaac Newton, Roger Cotes (16821716), and Colin Madaurin, but their investigations do not come within
the scope of this history. Robert Simson and Matthew Stewart (17171785) exerted themselves mainly to revive Greek ge
ometry. An Italian geometer, Giovanni Ceva (16471734) deserves mention here; a theorem in elementary geometry bears his name. He was an hydraulic engineer, and as such 1
was several times employed by the government of Mantua. His death took place during the siege of Mantua, in 1734. He ranks as a remarkable author in economics, being the first clearsighted mathematical writer on this subject. In 1678 he published in Milan a work, De lineis rectis. This contains " Ceva's Theorem " with one static and two geometric proofs Any three concurrent lines :
through the vertices of a triangle divide the opposite sides so that
^
a
=
Ca A/3 By Ba Oft Ay. In Ceva's book the properties of rec 
proved by considering the properties of the centre of inertia (gravity) of a system of points. 2 tilinear figures are
Modern Elementary Geometry
We
find it convenient to consider this subject
following four subheads: (2)
Modern Geometry 1
JButtetm ,
(1)
of the Triangle
Am. Math. &MX, Notes VL, VIL
under the
Modern Synthetic Geometry, and
Circle, (3)
Vol. 22, 1915, p. 100.
SYNTHETIC GEOMETBY Euclidean Geometry,
The
first
(4)
257
Textbooks on Elementary Geometry.
of these divisions has reference to
modern synthetic
methods of research, the second division refers to new theorems in elementary geometry, the third considers the modern conceptions of space and the several geometries resulting there from, the fourth discusses questions pertaining to geometrical teaching.
Modern
It was reserved for the Synthetic Geometry. of Gfaspard Monge (1746 genius 1818) to bring modern synthetic into the and to open up new avenues foreground, geometry I.
To avoid the long arithmetical computations in connection with plans of fortification, this gifted engineer sub stituted geometric methods and was thus led to the creation of of progress.
descriptive geometry as a distinct branch of science. Monge at the Normal School in Paris during the four
was professor months of its
existence, in
1795
;
he then became connected
with the newly established Polytechnic School, and later accompanied Napoleon on the Egyptian campaign. Among the pupils of Monge were Dupin, Servois, Brianchon, Hachette, Charles Julien Brianchon, born in Sevres Biot, and Poncelet. in 1785, deduced the theorem,
"Pascal's
Theorem" by means
known by
his
of Desargues'
name, from properties
of
what are now called polars. 1 Brianchon's theorem says " The hexagon formed by any six tangents to a conic has its opposite :
vertices connecting concurrently." The point of meeting is " sometimes called the " Brianchon point, Lazare Nicholas Marguerite Carnot (17531823) was born at
Kolay in Burgundy. At the breaking out of the Revolution he threw himself into politics, and when coalesced Europe, in 1793, launched against France a million soldiers, the gigantic i ,
Briancnon's proof appeared in "Memoirs sur les Surfaces courbes in Journal de VEcole Polgtechnique, T. VL, 297311,
da second Begre," 1806.
It is reproduced &
by TXTLOK,
op. tit., p. 290.
A HISTOKY OF MATHEMATICS
258
task of organizing fourteen armies to meet the enemy was achieved by him. He was banished in 1796 for opposing
Napoleon's coup d'etat. His Geometry of Position, 1803, and his Essay on Transversals, 1806, are important contributions to
modern plane geometry.
By his
effort to explain the
mean
" ing of the negative sign in geometry he established a geome " try of position which, however, is different from Von Staudt's work of the same name. He invented a class of general theo
rems on protective properties of figures, which have since been studied more extensively by Poncelet, Chasles, and others. Jean Victor Poncelet (17881867), a native of Metz, engaged in the Russian campaign, was abandoned as dead on the bloody field of Krasnoi, and from there taken as prisoner to Saratoff.
Deprived of books, and reduced to the remem
brance of what he had learned at the
Lyceum
at
Metz and
the Polytechnic School, he began to study mathematics from its elements. Like Bunyan, he produced in prison a famous
work, Trait& des Propriet&s projectwes des Figures, first pub Here he uses central projection, and gives
lished in 1822. the of "
theory
To him we owe the Law As an inde reciprocal polars.
reciprocal polars."
of Duality as a consequence of pendent principle it is due to Joseph Diaz Gergonne (1771We can here do no more than mention by name a few 1859).
the more recent investigators
Augustas Ferdinand Mobius Jacob Steiner 17901868), (17961863), Michel Chasles (1793Karl von Staudt (17981867). Chasles Christian 1880), Georg >f
:
Introduced the bad term anharmonic ratio, corresponding to the German Doppelverhattniss and to Clifford's more desirable crossratio.
Yon
and from metrical
Staudt cut loose from
all algebraic
relations, particularly the metrically
formulae
founded
and Chasles, and then created a geom which is a complete science in itself, inde
crossratio of Steiner
etry of position, pendent of all measurement.
259
ELEMENTARY GEOMETRY II.
Modem
Geometry of
and Oirde.
the Triangle
We
can
not give a full history of this subject, but we hope by our remarks to interest a larger circle of American readers in the and circle. 1 Fre recently developed properties of the triangle " nineis the quently quoted in recent elementary geometries be the middle let D, E, point circle." In the triangle ABO,
F
points of the sides, let AL,
BM, 'ON
be perpendicu lars to the sides, let a, b, e
be the middle points of
AO, BO, CO, then a cir cle can be made to pass through the points, L, D, c,
E, M,
circle is circle."
a,
N, F, b;
this

D
L
the "ninepoint By mistake, the earliest discovery of this circle
2 has been attributed to Euler.
There are several indepen dent discoverers. In England, Benjamin Bevan proposed in theorem Leybourn's Mathematical Repository, I., 18, 1804, a for proof
1
which
practically gives us
the ninepoint
A systematic treatise on this subject, which we
commend
circle.
to students
A. EMMERICH'S Die Brocardschen Geoilde, Berlin, 1891. Our historical notes are taken from this book and from the following papers : JULIUS LANGE, Geschichte des Feuerbachschen Rreises, Berlin, 1894; J. S, is
MACKAT, History of the
symmedian
point,
the ninepoint circle, pp. 1957, Early history of pp. 92104, in the Proceed, of the Edinburgh
See also MACKAT, The Wallace line Math, floe., Vol. XL, 189293. and the Wallace point in the same journal, Vol. IX., 1891, pp. 8391 E. LEMOINE'S paper in Association franchise pour Vavancement des VIGARIE, in the same publica Sciences, Congres de Grenoble, 1885 The progress in the geometry of the tri tion, Congres de Paris, 1889. in Progreso mat. L, 101106, angle is traced by VIGARIE for the year 1890 ;
;
.
de Math, elem., (4) 128134, 187190 for the year 1891 in Journ. Consult also CASET, Sequel to Euclid. ;
710, 3436. 2
MACKAT,
op.
cit.,
VoL XL,
p. 19.
1.
A HISTOEY OF MATHEMATICS
260
The proof was supplied to tlie Repository, Vol. I., Part 1 a problem, p. 143, by John Butterworth, who also proposed solved by himself and John Whittey, from the general tenor of which
knew
the circle in question These nine points are explic
appears that they to pass through all nine points. it
itly mentioned by Brianchon and Poncelet in Gergonne's In 1822, Karl Wtthelm Annales de Math&matiqiws of 1821. Feuerbach (18001834) professor at the gymnasium in Erlangen,
published a pamphlet in which he arrives at the ninepoint circle, and proves that it touches the incircle and the ex
The Germans
circles.
demonstrations of article
called
"Feuerbach/s Circle."
it
above referred
to.
The
last
this remarkable circle, so far as
Many
properties are given in the
its characteristic
independent discoverer of
known,
is
E. S. Davies, in an
1827 in the Philosophical Magazine, IL, 2931. In 1816 August Leopold Crelle (17801855), the founder of
article of
a mathematical journal bearing his name, published in Berlin a paper dealing with certain properties of plane triangles. He showed how to determine a point O inside a triangle, so that the angles (taken in the same order) formed by the lines joining it to the vertices are equal.
In the adjoining figure the three marked angles are equal. If the construction be
made
so as to
give angle Cl'AC
= &CB = &BA, then
a second point
O
t
is
obtained.
study of the properties of these
The new
angles and new points led Crelle to exclaim " It is indeed wonderful that :
so simple
a
figure as the triangle is so
How many
as yet unknown there not be!" Investiga tions were made also by 0. F. A* Jacobi of Pforta and some of Ms pupils, feo& after Ms death, in 1855, the whole matter in properties.
properties of otiher figures
may
ELEMENTAKY GEOMETRY
261
was forgotten. In 1875 the subject was again brought before the mathematical public by H. Brocard, who had taken up this study independently
a few years
The work
earlier.
of
Brocard was soon followed up by a large number of investi gators in France, England, and Germany. The new researches gave rise to an extended new vocabulary of technical terms.
Unfortunately, the names of geometricians which have been attached to certain remarkable points, lines, and circles are not
always the names of the men who first studied their properties. we speak of "Brocard points" and "Brocard angles/ 7 but
Thus,
C
historical research brought out the fact, in 1884,
and 1886^ that
these were the points and lines which had been studied by ? Crelle and C. F. A. JacobL The " Brocard Circle/' is Brocard s r own creation. In the triangle JLBC, let O and O be the first
and second "Brocard point" Let A* be the intersection of and OO; (?, of JBO r and Atl The BQ and CQ' J3', of is the "Brocard circle." circle passing through A\ B , ;
A&
f
A'B'C*
A
is
rf
B"G**
"Brocard's
is
O
first triangle."
Another like triangle,
called "Broeard's second triangle."
A*\
B
tihe
circumference of the "Brocard circle."
ff
,
f (7 together with C, O',
and two other
The
points
points, lie in
A HISTORY OF MATHEMATICS
262
In 1873 Emile Lemoine called attention to a particular point within a plane triangle which since has been variously called the "Lemoine point," "symmedian point," and "Grebe point." Since that time the properties of this point and of the lines
and
circles
To lead up figure, if
connected with
it
to its definition
CD is
so
drawn as
have been diligently investigated.
we premise that, in the adjoining to make angles a and 6 equal, then one of the two lines
AB and CD
the antiparallel of the other, with reference to the angle O. 1
is
Kow is
bisector of
AB,
tor of the antiparallel of
AB>
the symmedian
(ab
is
m&Mane). The in a triangle is
OE, the
the median and OF, the bisec
called
breviated from sym&trique de Za of the three symmedians concurrence of point the "
Tucker, symmedian point" out that some of the properties of this point, recently brought to light, were discovered previously to 1873. The antiparallels of a triangle which pass through called, after
Mackay has pointed
its
symmedian
point,
meet
its sides
in six points
which
lie
Lemoine circle." The "first Lemoine circle" is a special case of a "Tucker circle" and concentric with the "Brocard circle." The "Tucker circles" on a
circle, called the "second
maybe
thus defined.
Let DF*
= FE = ED' 1
; let, moreover, the following pairs of lines be antiparallels to each other:
The definition of antiparallels is attributed by EMMERICH (p. 13, note) The term antiparallel is defined by E. STONE in his New Mathem. Diet., London, 1743. Stone gives the above definition, refers to Leibniz in Acta Erudit., 1691, p. 279, and attributes to Leibniz a The word antiparallel is given in definition different from the above. MURRAY'S New English Dictionary of about 1660. See Jahrbuch fiber die Fvrtsehritte der Mdthemat^ Bd, XXTT. ? 1890, p. 46 ; Nature, XLL t 1
to Leibniz.
KH40&
ELEMEKTAET GEOMETRY
B and ED\ BO and FE', QA and DP ; D D\ E, E', F, F lie on a "Tucker
then the six points
1
circle."
,
}
263
Vary the
length of the equal antiparallels,
and the various cles"
"Tucker
cir
Allied to
are obtained.
these are the "Taylor circles." Still
different
"Neuberg kay
types
circles"
circles."
are
the
and "Mac
Perhaps enough
has been said to
call to
mind the
wonderful advance which has been made in the geometry of B' the triangle and circle during the latter half of the nineteenth century.
That new theorems
should have been found in recent time
the more remark
is
when we
consider that these figures were subjected to close examination by the keenminded Greeks and the long
able
line
of
geometers
who
1 p have since appeared.
Of
interest is the in
vention in 1853 by Sarrut of an apparatus for gener ating rectilinear motions it
has
overlooked. is
;
been altogether Celebrated
the invention in 1864
of an apparatus for con
verting circular into recti linear motion, by A. Peaucellier, a French army 2
officer.
Let
1 For a more detailed statement of researches in England see article " The Kecent General Report of Fourteenth the of Triangle," Geometry A. I. G. Tn 1888, pp. 3548. 2 PEATJCELLIEK'S articles appeared in N'ouvelles Annales, 1864 and 1873
A HISTOBY OF MATHEMATICS
264
ACBP be a rhombus
whose sides are
less than the equal sides the straight lines which are drawn in full to be bars or "links," jointed at the points A, B,
of the angle
AOB. Imagine
P
will describe Make C describe the circle and C, P, Q, 0. Eemove CQ, fixed. the straight line PD, the pivot being will trace the let C move along any curve in the plane, then
P
inverse of that curve with respect to 0. Peaucellier's 1 of linkages was developed further by Sylvester.
method
we have spoken of the instruments used in how the Greeks used the ruler and compasses, and how later a ruler with a fixed opening Elsewhere
geometrical constructions
came
of the compasses, or the ruler alone,
few geometers.
Of
to be used
interest in this connection is
by a a work by
the Italian Lorenzo Mascheroni (17501800), entitled Geom&tria del compasso, 1797, in which all constructions are made
with a pair of compasses, but without restriction to a fixed radius. The book was written for the practical mechanic, the
author claiming that constructions with compasses are 2 The work secured the ruler.
more accurate than those with a
who proposed to the .French mathematicians the following problem To divide the circum ference of a circle into four equal parts by the compasses only. This construction is as follows Apply the radius three times attention of Napoleon Bonaparte,
:
:
to the circumference, giving the arcs
distance
AD is
a diameter.
With a
AB, BQ, CD.
Then the
radius equal to
AC, and
1
SYLVESTEK in Educ. Times Beprint, Vol. XXL, 68, (1874). See C. TAYLOB, Ancient and Modern Geometry of Conies, 1881, p. LXXXVTL Consult also A. B. KEMPE, How to draw a straight line, a lecture on Unkages, London, 1877. 2 See CHARLES HUTTON'S Phttos. and Math. Diet., London, 1815, " article " Geometry of the Compasses MAEIE, X. p, 98 ; KLEIN, p. 26 ; ST&INBB, Gtesammelte WerJce, L, 463 ; Mascneroni's book was translated ;
into 3?reBeh in 1798
German in 1880.
;
an abridgment of
it
by HFTT was brought out
in
ELEMENTARY GEOMETER
265
A
and D, draw arcs intersecting at E. Then EO, the centre of the given circle, is the chord of the quadrant of the cijcle. The inscription of the regular polygon of 17 sides was first with centres
where
O
is
by Carl Friedrich Gauss (17771855), when a boy of March 30, 1796. At that time he was undecided whether to choose ancient lan guages or mathematics as his specialty. His success in this effected
nineteen, at the University of Gottingen,
1 inscription led to the decision in favour of mathematics.
A
mode of construction has been suggested inde pendently by a German and a Hindu. Constructions are to be effected by the folding of paper. Hermann Wiener, in 1893, curious
showed how to construct, by folding, the nets of the regular In the same year, Sundara Row published a little book " On paper folding " (Addison & Co., Madras), in which it is shown how to construct any number of points on the solids.
2 ellipse, cissoid, etc.
In connection with polyedra the theorem that the number of edges falls short by two of the combined number of vertices
and faces is interesting. The theorem is usually ascribed to 3 It is true Euler, but was worked out earlier by Descartes. only for polyedra whose faces have each but one boundary j if a cube
is
placed upon a larger cube, then the upper face of the
larger cube has two boundaries, an inner and an outer, and the theorem is not true. A more geneiral theorem is due to
F. 1
Other modes of inscription of the 17sidecL polygon were given by STAUDT, in Crelle, 24 (1842), by SCHKOTJBK, in CreUe, 7& (1872). The latter uses a ruler and a single opening of the compasses. By $ie compasses only, the inscription has also been effected. See'K&Knr,
VON
27 ; an inscription Leipzig, 1872, p. 67. p.
*
is
given also in
PAUL BACHMANN, Ereistheibmg, 2
KLEIN, p. 33L, See E. BE JoNQui&ftEs in Bibliotk. Mathem., 1890, $. 43. " mr Theorie der Polyeder " Site.Ber d. Wien. LIFPICH, Akad^ Bd.
A HISTORY OP MATHEMATICS
266
HI. NonEudidean Geometry. The history of this subject centres almost wholly in the theory of parallel lines. Prelimi nary to the discussion of the nonEuclidean geometries, it may be profitable to consider the various efforts towards simplifying and improving the paralleltheory made, (1) by giving a new defini tion of parallel lines, or by assuming a new postulate, different
from Euclid's postulate
parallelpostulate, (2) by proving the parallelof the straight line and plane angle.
from the nature
Euclid's definition, parallel straight lines are such as are in
same plane, and which being produced ever so far both ways
the
do not meet,
still
holds
its
place as the best definition for use in
Anew
definition, used by Posidonius was adopted by the painter, Albrecht Durer. He wrote a geometry, first printed in 1525, in which parallel lines
elementary geometry. (1st Cent. B.C.),
A
are lines everywhere equally distant. 1 little later Clawus, in his edition of Euclid of 1574, in a remark, assumes that a line
which
is
everywhere equidistant from a straight line is it This postulate lies hidden in the definition of
self straight.
The
IMlrer.
objection to this definition or postulate
is
that
an advanced theorem, involving the difficult consider ation of measurement, embracing the whole theory of incom
it is
mensurables.
Moreover, in a more general view of geometry Clavius's treatment is adopted by
must be abandoned. 2
it
Jacques Peletier
(15171582)
of
Paris,
Boscomch (17111787),3 Johann Christian
Euggiero
Giuseppe
Wolf (16791754)
see also H. DiraiGE, Theorie der FunJtiionen, Leipzig, 1882, 84, 1881 p. 226 ; an English translation of this work, made by G. E. FISHER and ;
L
SCHWATT, has appeared. GITNTHEE, Math. Unt. im d. Mittela., pp. 361, 362. 2 See LOBATCHEWSKY, The Theory of Parallels, transl. by G. B. HALSTED, Austin, 1891, p. 21, where the theorem is proved that in " The farther lines are on the J, 1
S.
pseadoHspherical space, parallel prolonged side of their parallelism, the more they approach one another." He was pax>lessor at several Italian universities, was employed in
ELEMENTARF GEOMETRY
267
Simpson (in the first edition, of Ms Elements, John Bonnycastle (1750 ?1821) of the Royal Military 1747), at Woolwich, and others. It has been used by a few Academy 1 authors. This definition is the first of several given American of Halle, TJiomas
by E. Stone in his New Mathematical Dictionary, London, 1743 ; in fact, "in the majority of textbooks on elementary geome try, from the sixteenth to the beginning of the eighteenth lines
century, parallel
are
declared
to be
lines
that are
be sure, very convenient." 2 But is, equidistant objections to this mode of treatment soon arose. As early as 1680 Giordano da Bitonto, in Italy, said that it is inadmissible,
which
to
unless the actual existence of equidistant straight lines can be established. Saccheri rejects the assumption unceremoni 3
ously.
Another
definition,
which involves a
make
tacit assumption,
de
same angle with a third line. The definition appears to have originated in France; it is given by Pierre Varignon (16541722) and Etienne Bezout (17301783), both of Paris. In England it was used by Gooley,4 in the United States by H. "N. Robinson. clares parallels to be lines which
A
slight
modification
of this
is
Parallels are lines perpendicular to
a
the
the following definition: This is recom
third line.
by several popes, was in London in 1762, and the Royal Society as a proper person to be ap pointed to observe the transit of Venns at California.^ but the suppression of the Jesuit order, which he had entered, prevented his acceptance of various scientific duties
was recommended by
the appointment. See Penny Cyclopaedia. 1 Consult Teach, and Hi&t. of Math, in the U. 2 8
ENGEL and STACKEL, ENGEL and STACKEL, " MINOS
&,
p, 377,
p. 33. p. 46.
So far as I can make out Mr. Cooley quietly assumes that a pair of lines which make equal angles with one line, do so with &U lines. He might just as well say that a young lady, who was inclined to one young man, was equally and similarly inclined to all young men 1 RHJLDAMJLJTTHDS ; She might make equal angling ' with them all any*
:
'
1
4
A HISTORY OP MATHEMATICS
268
mended by the
Italian
G.
A. Borelli in 1658, and by the cele
brated French textbook writer, 8. F. Lacroix. 1 Famous is the definition, parallel lines are straight lines ing the
same
direction.
At
Jiav
first, this definition attracts us
by But the more it is studied, the more perplexing are the questions to which it gives rise. Strangely enough, authors adopting this definition encounter no further diffi culty with parallel lines nowhere do they meet the necessity its simplicity.
;
of assuming Euclid's parallelpostulate or any equivalent of
A
question which has perplexed geometers for centuries In actual fact, there is, appears disposed of in a trice!
it!
perhaps, no
word
in mathematics which,
by its apparent sim and obscurity, has misled so plicity, but real indefiniteness many able minds. In the United States, as elsewhere, this definition has been widely used, but for the sake of sound 2 learning, it should be banished from textbooks forever.
A new definition of parallel lines, suggested by the principle of continuity,
and one
of great assistance in advanced
try, though unsuited for elementary instruction,
how.' 1
M Ed,,
C. L. DOIXJSON, Euclid p.
2
;
and Sis Modern
is
geome
that first
Rivals^ London, 1885,
also p. 62.
1 S. 3?. LACBOIX, Essais sur V enseignement en general, et sur celui des mathematiques en particulier, Paris, 1805, p. 317. 2 The the term "direction" have been ably set forth objections t so often, that we need only refer to some of the articles: DE MOBGAH'S review of J. M. Wilson's Geometry, Mhenamm, July 18, 1868 ; E. L. RICHARDS in Educational Review, Vol. III., 1892, p. 32 ; Seventh General Report (1891) of the A. I. G. T., pp. 3642 ; G. B. HALSTED, The Science Absolute of Space by JOHN BOLTAI, 4th Ed., 1896, pp. 6371; G. B. HALSTED in Educational Beview, Sept., 1893, p. 153; C. L. DODGSON, Euclid and His Modern RfooZs, London H. MBLLEB, Bestizt die hevMge Schulgeometrie noch die Vorznge des EuTdidischen Originals f Metz, 1889, p. 7 ; Zeitschrift f&r mathematischen und naturwissenschafttiGhen Teach, and Hist, of Math. UnterricU, Tenbner in Leipzig, XVI., p. 407 the U. 8., pp. 381383; Monist, 1892 (Review of E. T. Drtow's ;
;
m
of Geometry").
ELEMENTARY GEOMETRY
269
given by John Kepler (1604) and Girard Desargues (1639): Lines are parallel if they have the same infinitely distant similar idea is expressed by E. Stone in point in common. his dictionary thus " If be a point without a given indefi nite right line CD ; the shortest line, as AB, that can be drawn
A
:
from
A
to
parallel to
it, is
A
perpendicular
;
and the longest, as EA,
is
CD."
Quite a number of substitutes for Euclid's parallelpostulate differing in form, but not in essence, have been suggested
In the evening of July 11, 1663, John WaUis delivered a lecture at Oxford on the parallelpostulate. 1
at various times.
He recommends in place of Euclid's postulate To any triangle another triangle, as large as you please, can be drawn, which is similar to the given triangle. Saccheri showed that Euclidean :
geometry can be rigidly developed
if
the existence of one
tri
Lam angle, unequal but similar to another, be presupposed. Wallis's postulate was again bert makes similar remarks. proposed for adoption by L. Carnot and Laplace, and more by J. Delboeuf? Alexis Claude Clairaut (17131765), a famous French mathematician, wrote an elementary geom etry, in which he assumes the existence of a rectangle, and from recently
this substitute for Euclid's postulate develops the elementary
theorems with great clearness. Other equivalent postulates are the following: a circle can be passed through any three points not in the same straight line (due to W. Bolyai) ; the existence of a finite triangle whose anglesum is two right angles (due to Legendre) ; through every point within an angle
a line can be drawn intersecting both sides (due to J. 3?. Lorenz (1791), and Legendre) in any circle, the inscribed equi ;
lateral quadrangle is greater
than
aij$r
one of the segments
>*.
1
2
See Opera, Vol. H, 665678. and SS&CKEL, pp. 2130. EWGEI, and STACKUL, p. 19.
In German translation
it is
given ty
A HISTORY OF MATHEMATICS
270
which
lie outside of ic (due to C. L. Dodgson); two straight which intersect each other cannot be both parallel to the same straight line (used by John Playfair). 1 Of all
lines
met with general edition of Euclid, and
these substitutes, only the last has
Playfair adopted
it
in his
favour. it
has
been generally recognized as simpler than Euclid's parallelpostulate.
Until the close of the
first
quarter of the nineteenth century
was widely believed by mathematicians that Euclid's parallelpostulate could be proved from the other assumptions and the it
definitions of geometry.
We
have already referred to such We refrain from
made by Ptolemy and Nasir Eddin. discussing proofs in detail. They all fail,
efforts
either because an
equivalent assumption is implicitly or explicitly made, or because the reasoning is otherwise fallacious. On this slippery ground good and bad mathematicians alike have fallen. We are told that the great Lagrange, noticing that the formulae of spherical trigonometry are not dependent upon the parallelpostulate,
hoped to frame a proof on this fact. Toward the he wrote a paper on parallel lines and began
close of his life
to read it before the said: "II faut
again)
Academy, but suddenly stopped and 2 songe encore" (I must think it over
que j'y he put the paper in his pocket and never afterwards
;
publicly recurred to
it.
The
researches of Adrien Marie Legendre (17521833) are interesting. Perceiving that Euclid's postulate is equivalent
theorem that the anglesum of a triangle is two right angles, he gave this an analytical proof, which, however, assumes the existence of similar figures. Legendre was not to the
satisfied
with
this.
Later, he proved satisfactorily by assum
ing lines to be of indefinite extent, that the 1
Consult G. B. HALSTED'S Bolyai^
*
ENOBL and STACKED,
p. 212.
4th.
anglesum can
Ed., pp. 64, 66.
ELEMENTARY GEOMETRY
271
not exceed two right angles, but could not prove that it cannot two right angles. In 1823 3 in the twelfth edition
fall short of
he thought he had a proof for Afterwards, however, he perceived its weak rested on the new assumption that through any
of his Elements of Geometry,
the second part. ness, for it
point within an angle a line can be drawn, cutting both sides of the angle. In 1833 he published his last paper on parallels, in which he correctly proves that, if there be any triangle
sum of whose angles is two right angles, then the same must be true of all triangles. But in the next step, to show the
rigorously the actual existence of such a triangle, Ms demon stration failed, though he himself thought he had finally settled
the whole question. 1 quite so far as had
As a matter
of fact
he had not gotten
2 years earlier. Moreover, before the publication of his last paper, a Eussian mathematician had taken a step which far transcends in
Saccheri one hundred
boldness and importance anything Legendre had done on this subject.
As with the problem of the quadrature of the circle, so with the parallelpostulate: after numberless failures on the part of some of the best minds to resolve the difficulty, certain shrewd thinkers began to suspect that the postulate 1
A novel attempt to prove the anglesum of
a triangle to be
two
right
angles was made in 1813 by John Playf air, in his Elements of Geometry. See the edition of 1855, Philadelphia, pp. 295, 296. It consists, briefly, in laying a straight edge along one of the sides of the figure, turning the edge round so as to coincide with each in turn.
and then Then the
is said to have described angles whose sum is four right angles. The same argument proves the anglesum of spherical triangles to be two right angles, a result which we know to be wrong. It is *to be regretted
edge
that in two American textbooks, just published, this argument is repro duced, thus giving this famous heresy new life. For the exposition of
the fallacy see G. B. EJLLSTED'S 4th Ed. of BOLTAI'S Science Absolute of Space, pp. 6371 ; Nature, Vol. XXIX., 1884, p. 453, 2
ENGEL
aM STACKBL, pp. 212, 213,
'
A HISTORY OF MATHEMATICS
272
This scepticism appears in various
did not admit of proof. 1
writings. If it required courage
on the part of Euclid to place his
parallelpostulate, so decidedly unaxiomatic,
among
his other
was de postulates and common notions, even greater courage manded to discard a postulate which for two thousand years
had been the main corner stone of the geometric structure. Yet several thinkers of the eighteenth and nineteenth cen turies displayed that independence of thought, so essential to
great discoveries.
While Legendre
still
endeavoured to establish the parallel
postulate by rigorous proof, Nicholaus Ivanovttch Lobatchewsky (17931856) brought out a publication which assumed the
contradictory of that axiom. Lobatchewsky's views on the foundation of geometry were first made public in a discourse before the physical and mathematical faculty of the University of Kasan (of which he was then rector), and first printed in
Kasan Messenger
the
for 1829,
and then
in
the
Gelehrte
Sdiriflen der Uhwersitat Kasan, 18361838, under the title "!N"sw Elements of Geometry, with a complete theory of parallels."
This work, in Eussian, not only remained un
known
to foreigners, but attracted no notice at home. In 1840 he published in Berlin a brief statement of his researches.
The
distinguishing feature of this "Imaginary Geometry" of Lobatchewsky is that through a point an indefinite number of lines can be drawn in a plane, none of which cut a given line in the plane, is variable,
2
and that the sum
though always
less
of the angles in a triangle
than two right angles. 2
and STACKEL, pp. 140, 141, 213215. LOBATCHEWSKY'S Theory of Parallels has
"been
translated
into
English by G. B. HALSTED, Austin, 1891. It covers only forty pages. G. B. Halsted has also translated Professor A. VASILIEV'S Address on tfese life and researches of Lobatchewsky, Austin, 1894.
273
ELBMENTABY GEOMETRY
A similar system of geometry was devised
by the Bolyais in " absolute them and called geometry." Wolfgang Hungary by Bolyai de Bolya (17751856) studied at Jena, then at Gottingen, where he became intimate with young Gauss. Later, he be came professor at the Reformed College of MarosVasa*rhely, where for fortyseven years he had for his pupils most of the Glad in old time planter's he was truly original in his private life as well as in his mode of thinking. He was extremely modest. ISTo monument, said he, should stand over his grave, only an appletree, in memory of the three apples the two of Eve and Paris, which
present professors in Transylvania. garb,
:
made
and that of Kewton, which elevated 1 His son, the earth again into the circle of heavenly bodies. Johann Bolyai (18021860) was educated for the army, and distinguished himself as a profound mathematician, an im He once passioned violinplayer, and an expert fencer. officers on that of thirteen condition the challenge accepted after each duel he might play a piece on his violin, and he hell out of earth,
2 vanquished them all
"Wolfgang BolyaFs chief mathematical work, the Tentamen, The first volume is appeared in two volumes, 18321833. followed by an appendix written by his son Johann on The Science Absolute of Space. Its twentysix pages make the
name
of
Johann Bolyai immortal
Yet
for thirtysix years remained in
this appendix, as also Lobatchewsky's researches,
almost entire oblivion.
Finally Bdchard Baltzer, of Giessen,
in 1867, called the attention of mathematicians to these

*. SCHMIDT, "Aus dem Leben zweir nngarischer Johann tmd Wolfgang Bolyai von Bolya," GBUKBKT'S Archfo der Mctfhematik uwd Pfa/sifc, 48 2, 1868. 2 For additional "biographical detail, see G. B. HAUSTED'S translation of JOHN BOLTAI'S The Science Absolute of Space, 4th EcL 1896. Dr. HaJsted is about to issue The Life of Bolyai, containing the Autobiog raphy of Bolyai Farkas, and other interesting information. :
A
274 derful studies.
HISTORY OF MATHEMATICS In 1894 a monumental stone was placed on
Johann Bolyai in MarosVasarhely. In the years 18931895 a Lobatchewsky fund was secured the longneglected grave of
through contributions of scientific
men
in all countries,
which
goes partly toward founding an international prize for re search in geometry, and partly towards erecting a bust of Lobatchewsky in the park in front of the university building in Kasan,
But the Eussian and Hungarian mathematicians were not the only ones to
When
Gauss
whom
the
new geometry suggested
saw the Tentamen
of the elder Bolyai, his
itself.
former
roommate at Gottingen, he was surprised to find worked out there what he himself had begun long before, only to leave it still
him
in his papers. His letters show that in 1799 he was trying to prove a priori the reality of Euclid's system, but
after
he became convinced that this was impossible. Many writers, especially the Germans, assume that both Lobatchew
later
sky and Bolyai were influenced and encouraged by Gauss, but no proof of this opinion has yet been presented. 1 of
Becent historical investigation has shown that the theories Lobatchewsky and Bolyai were, in part, anticipated by two
writers of the eighteenth century, Geronimo Saccheri (16671733), a Jesuit father of Milan, and Johann Heinrich Lambert
Both made (17281777), a native of Miihlhausen, Alsace. the of three of space, definitions kinds researches containing now called the nonJEuclidean, Spherical, and the Euclidean 2
geometries. 1
EKGEI/ and STXCKEL, pp. 242, 243 ; G. B. HJLLSTED, in
/Science, Sept.
6, 1895. 2
The researches of Wallis, Saccheri, and Lambert, together with a full history of the parallel theory down to Gauss, are given "by ENGEL and STACKEL. See also G. B. HALSTEI>'S "The nonEuclidean Geometry in evitable" in the Monist, July, 1804. Space does not permit us to speak of the later history of nonEuclidean geometry. commend to our
We
ELEMENTARY GEOMETRY
We have touched upon
275
the subject of nonEuclidean geome
try, because of the great light which these studies have thrown upon the foundations of geometric theory. Thanks
to these researches,
no intelligent author of elementary text
books will now attempt to do what used to be attempted: prove the parallelpostulate. We know at last that such an attempt
is futile.
Moreover,
many
easily recognized as fallacious
trains of reasoning are
by one
who
now
sees with the eyes
of Lobatchewsky and Bolyai. Possessing, as we do, English of their epochmaking works, no progressive teacher of geometry in High School or College certainly no translations
author of a textbook on geometry
can afford to be ignorant
of these results.
IY. Textbooks on Elementary Geometry. The history of the evolution of geometric textbooks proceeds along different lines in the various European countries. About the time when
Euclid was translated from the Arabic into Latin, such intense veneration came to be felt for the book that it was considered
modify anything therein. Still more pronounced was this feeling toward Aristotle. Thus we read of Petrus sacrilegious to
Ramus
(15151572) in France being forbidden on pain of corporal punishment to teach or write against Aristotle. This royal mandate induced Bamus to devote himself to mathetranslations of Lobatchewsky and Bolyai. Georg Bernhard Riemann's wonderful dissertation of 1864, entitled Ueber die Hypothesen welche der Greometrie zu Gfrunde liegen is trans lated by W. K. Cliford in Nature, Vol. 8, pp. 1417, 3637. Eiemann developed the notion of nply extended magnitude. Helmholtz's lecture, entitled "Origin and Significance of Geometrical Axioms" is contained in a volume of Popular Lectures on Scientific Subjects, translated by E. ATKINSON, New York, 1881. Consult also the works of W. K. Clifford, The bibliography of nonEuclidean geometry and hyperspace is given by G. B, HALSTED in the American Journal of Mathematics, Vols. I. and II. Readers may be interested in Flatland, a Romance of Many Dimensions,
readers
Halsted's
3?riedrich
published by Roberts Brothers, Boston, 1885.
A HISTORY OF MATHEMATICS
276 lie
matics;
brought out an edition of Euclid, and here again He did not favour inves
displayed his bold independence. tigations
on the foundations of geometry
was not
at all desirable to carry everything back to
;
he believed that
axioms; whatever is evident in itself, needs no proof. opinion on mathematical questions carried great weight.
it
a few
His
His
views respecting the basis of geometry controlled French textbooks down to the nineteenth century. In no other civil ized country has Euclid been so little respected as in France. conspicuous example of French opinion on this matter is
A
the text prepared by Alexis Claude Clairaut in 1741. He con demns the profusion of selfevident propositions, saying in his " It is not that Euclid should himself preface,
give
surprising
the trouble to demonstrate that two circles which intersect
iave not the same centre; that a triangle situated within another has the sum of its sides smaller than that of the sides
which contains
of the triangle
vince obstinate
is
That geometer had to con gloried in denying the most
it.
but in our day things have changed face ; reasoning about what mere good sense decides in advance now a pure waste of time and fitted only to obscure the
evident truths all
sophists,
who
,
.
.
;
truth and to disgust the reader." This book, precisely antip odal to Euclid, has contributed much toward moulding opin ions on geometric teaching, but otherwise it did not enjoy a 1 great success.
Similar views were entertained
by Etienne
Bzout
(17301783), whose geometric works, like Clairaut's, are deficient in rigour. Somewhat more methodical was
But the French Sylvestre Frangois Lacroix (17651843). geometry which enjoyed the most pronounced success, both at home and in other countries, is that of Adrien Marie e, first 1
brought out in 1794
It is interesting to note
See article " Geometric " in tbe Grand Dictionnaire Universal Siecle par PIEHEE LAKOUSSB.
ELEMENTARY GEOMETBY Loria's estimate of Legendre's Elements de
277
gomtrie.
1
"
They
[success], since, as regards form, if they can com with in clearness and precision of style, they are Euclid pete
deserved
it
superior to
him
in the complex
harmony which
gives
them the
appearance of a beautiful edifice, divided into two symmetrical parts, assigned, the one to the geometry of the plane and the other to the geometry of space ; and since in regard to matter, they surpass Euclid by being richer in material and better in
But the great French analyst, in writing of geometry, could not forget his own favoured studies, so that in his hands geometry became a vassal of arithmetic, from
certain particulars.
which he borrowed a few ratiocinations and even some of his nomenclature, and took from
its
dominions the whole theory
of proportions. If it is added that, while Euclid avoids the use of any figure, the construction of which is unknown to
the reader, Legendre uses without scruple the socalled f hypo thetical constructions/ and that he gave the preference to that
unfortunate definition of a straight line [used, moreover, even by Kant] as a minimum line there is sufficient argument to ;
support the fact that Legendre's edifice was not long in show * ing itself of a solidity incomparably inferior to its beauty." Thus we see that French writers, influenced by their views respecting methods of teaching, by their belief that what is
apparent to the eye may be accepted by the young student without proof, and by their general desire to make geometry easier and more palatable, allowed themselves to depart a long
way from the practice of Euclid. But Legendre is more strict than Clairaut; the reactioa against Clairaut's views became more and more pronounced, and about the middle of the nine1
LOBIA, Delia variafortuna di EncOde, Roma, 1893, p. 10. of Brewster's translation of Legendre's geom etry "was bimtgnt out in 1828 by Charles Davies of West Point. John Fairar of Harvard CoEege issued a new translation in 1830. 2
An AmediCan edition
A HISTOBY OF MATHEMATICS
278
teenth century, Jean Marie Constant Duhamel (17971872) and
Hoiid l began to institute comparisons between Legendrian and Euclidean methods in favour of the latter, recommending the return to a modified Euclid. Duhamel proclaimed the method of limits to be the only rigorous one by which the use of
J.
2 the infinite can be introduced into geometry, and contributed much toward imparting to that method a clear and unobjection able form. Hotiel, professor at Bordeaux, while expressing
his regret that Euclid's Elements
had
fallen into disuse in
Prance, did not recommend the adoption of Euclid unmodified. While DuhameFs and Hotiel's desired return to Euclidean
methods did not take
place, the discussion, nevertheless, led
The works which in to improvements in geometric texts. France now enjoy the greatest favour are those of E. RoucTie and
C. de Comberousse,
and
of Oh. Vacquant.
8
The treatment
by the method of limits, against which the cry "lack of rigour" cannot be raised. In appendices considerable attention is given to projective geometry, but of incommensurables is
Loria remarks that the old and the
new in geometry
are here
not united into a coherent whole, but merely mixed in dis 4
jointed fashion. Italy, the land where once Euclid
was held in high venera
where Saccheri wrote, in 1733, Eudides ab omni naevo wndicatus (Euclid vindicated from every flaw), finally discarded tion,
her Euclid and departed from the spirit of his Element*. In 1867 Professors Cremona of Milan and Battaglini of Naples 1 DTTHAMEL, Des methodes dans les sciences de raisonnement (five vol umes), 18561872, IL, 326; HOUEL, Essai sur les principes fondament aux de la geometrie elementaire ou commentaire sur les XJO77. premi eres propositions des elements ffEudide (Paris, 1867).
2
LORIA,
*
An
op. eft., p. 11.
8
LORIA, op.
cit., p. 12.
elementary geometry of great merit, modelled after Trench, works, particularly that of Rouche' and Comberousse, was prepared in 1870 by William Cliauvenet of Washington Uniyersity in St. Louis.
ELEMENTARY GEOMETRY
279
were members of a special government commission to inquire into the state of geometrical teaching in Italy. They found it
to be so unsatisfactory
and the number of bad textbooks so
much on
the increase, that they recommended for classical schools the adoption of Euclid pure and simple, even 1 This recom though Cremona admitted that Euclid is faulty. great and
so
mendation became law.
Later the use of Euclid's text was
2 replaced by that of other works, prepared on analogous plans. Thereupon appeared after the amended Euclidean model, as
distinguished from the Legendrian, the meritorious work by Later came the Elementi di Ovidio.
A. Sannia and E.
D
7
geometria of Biccardo de Paolis and the Fondamenti di geometria of Guiseppe Veronese.
The high esteem in which Euclid was held in Germany during early times is demonstrated by many facts. About the close of the eighteenth century Abraham Gotthelf Kastner remarked that "the more recent works on geometry depart from Euclid, the more they lose in clearness and thorough ness." This points to a falling off: from the Euclidean model.
The most popular texts in use about the middle of the nine teenth century, and even those of the present time, are, from a scientific point of view, anything but satisfactory. Thus in H. B. Lubsen's JZttementarGeometrie, 14th Ed., Leipzig, 1870, written in the country which produced Gauss, and at
a time when Lobatchewsky's and had been published 41 and 37 years
Bolyai's immortal works respectively,
we
still find
s If we bear in mind (p. 52) a proof of the parallelpostulate that geometry deals with continuous magnitudes, in which, com!
1
2
HIRST, address in First Annual Report, A. LofiiA, op.
L
G. T., 1871.
cit>, p. 15.
8
Another illustration of the confusion which prevailed regarding the foundations of geometry, long after Lobatchewsky, is found in Thomas Peronnet Thompson's Geometry without Axioms, 3d Ed., 1830, in which rid " of axioms and be endeavoured to " get
postulates
1
A HISTORY OF MATHEMATICS
280 mensurability
sen should,
is exceptional, it is
make no mention
work which has been used
somewhat
startling tliat Llib
Another Karl Koppe's
of incommensurables.
for several decennia
Planimetrie, 4th Ed., Essen, 1852.
is
It speaks of parallels as
having the same direction, and fails to consider incommen surability excepting once in a footnote. Kambly's inferior work appeared in 1884 in the 74th edition. 1 The subject of geometric teaching has been
Some
much
discussed in
Germany.
excellent textbooks have been written, but they do not
appear to be popular.
Among
the better works are those
H. Muller, Kruse, Worpitzky, Henrici Most of these seem to be dominated by Euclidean methods. 3 One of the questions debated in Ger many and elsewhere is that pertaining to the rigidity of fig ures. While Euclid sometimes moves a figure as a whole, he never permits its parts to move relatively to one another. With him, figures are " rigid." In many modern works this practice is abandoned. For instance, we often allow an angle of Baltzer, Schlegel,
and Treutlein. 2
to be generated
by
rotation of a line, 4
whereby we arrive
at
the notion of a " straight angle " (not used by Euclid), which now competes with the right angle as a unit for angular The abandonment of rigidity brings us in closer measure. is, we think, to be we proceed with circumspection.
touch with the modern geometry, and
recommended, provided that
Pure projective geometry needs neither motion nor
rigidity.
1 A. ZIWET in Bulletin N. T. Math. Ziwet here re 8oc., 1891, p. 6. views a book containing much information on the movement in Germany, viz. H. SCHOTTEN, Inkalt und Methode des planimetrischen Unterrichte. The reader will profit by consulting HOFFMANN'S Zeitschrift fur mccthematischen und nctfurwissenschqftlichen Unterricht, published by Teub
2
LOBIA, p. 19. SCHOTTBN, op. tit. ; H. MULLER, Besitet die hewtige &M&eometsrie noch die Vorztye des E&kltdiscken Originals f Metz, 1889. * H. MflLLEK, O^ C&., p. 2.
Ber, Leipzig. *
See LOELL,
p. 19,
ELEMENTARY GEOMETEY
281
Preparatory courses in intuitive geometry, in connection with geometric drawing, hare been widely recommended and adopted in
Germany. England has been the home of conservatism in geometric teaching. Wherever in England geometry has been taught, Euclid has been held in high esteem. It appears that the first to rescue the Elements from the Moors and to bring it out
The
Latin translation was an Englishman.
in
first
English
translation (Billingsley's) was printed in 1570. Before this, Robert Becorde made a translation, but it was probably never 1
Elsewhere we have spoken of mediaeval geomet teaching at Oxford. It was carried on with very limited
published. rical
success.
of
Merton
About 1620 Sir Henry Savile (15491622), warden College, endeavoured to create an interest in mathe
course of lectures on Greek These were geometry. published in 1621. On concluding the course he used the following language " By the grace of
matical studies by giving a
:
God, gentlemen hearers, have redeemed my pledge.
I have
performed
my
promise; I
I have explained, according to
ability, the definitions, postulates, axioms,
and the
my
first eight
Here, sinking under propositions of the Elements of Euclid. the weight of years, I lay down my art and, my instruments," 2
must be remembered that at this time scholastic learning and polemical divinity held sway. Savile ranks among those It
who
laboured for the revival of true knowledge.
He
founded
two professorships at Oxford, one of geometry and the other of astronomy, and endowed each with a salary of 150 per annum. In the preamble to the deed by which he annexed this salary he says that
and unknown i
W. W. B. BJLLL,
* Translated
geometry wasalmost totally abandoned
in England. Matiis. at
Cambridge, p. 18.
from the Latin by W. WHEWEIX, in
Induct. Bctences,
VoL
L,
New York,
1858, p. 206.
his Hist* of ike
A
282
HISTORY OF MATHEMATICS
Savile delivered thirteen lectures in his course, in which philological
and
historical questions received
more attention
In one place he says, "In the beautiful structure of geometry there are two blemishes, two " defects I know no more." These " blemishes are the theory At that time of parallels and the theory of proportion. Euclid's parallelpostulate was objected to as unaxiomatic
than did geometry
itself.
1
;
and requiring demonstration. We now know by the light de rived from nonEuclidean geometry that it is a pure as sumption, incapable of proof; that a geometry exists which assumes
its
contradictory; that
it
separates the Euclidean
from the pseudospherical geometry. No "blemish," there fore, exists on this point in Euclid's Elements.* The second "blemish" referred to the sixth statement in Book V. 2 Fedar gogically, this fifth
book
is still criticised,
on account of
its
young minds; bnt scientifically from certain unimportant emendations which Robert (aside Simson thought necessary) this book is now regarded as one
excessive abstruseness for
of exceptional merit. During the seventeenth and eighteenth centuries a consider able number of English editions of Euclid were brought out
These were supplanted after 1756 by Eobert Simson's EudicL In the early part of the nineteenth century Euclid was still without a rival in Great Britain, but the desirability of modi As early as 1795, John Playfication arose in some minds. brought out at Edinburgh his Elements of Geometry, containing the first six books of Euclid and a supplement, embracing an approximate quadrature of the
fair
circle
(17481819)
(i.e.
the computation of
etry drawn from other
IT)
and a book on solid geom fifth book of Euclid,
In the
sources.
Playfair endeavours to remove Euclid's diffriseness of style by
,
II. ,
364.
2
EKGEL and STACKEL,
p. 47.
28S
ELEMENTARY GEOMETRY
Playfair also introduces
introducing the language of algebra.
Euclid's. Gradually a parallelpostulate, simpler than Euclid's from deviations of more the advantage pronounced defects out De In 1849 text was expressed. Morgan pointed 2 1 and Jones 8 Wilson later About twenty years in Euclid.
new
Euclidean method. expressed the desirability of abandoning the in mathematicians Finally in 1869 one of the two greatest J. <J. Syl England raised his powerful voice against Euclid. vester said, "I should rejoice to see ... Euclid honourably shelved or buried deeper than e'er plummet sounded out '
s
of the schoolboy's reach."
*
These attacks on Euclid as a school
book brought about no immediate change in geometrical teach The occasional use of Brewster's translation of Legendre ing. or of Wilson's geometry no more indicated a departure from Euclid than did the occasional use, in the eighteenth century, 1
LORIA, op.
British
cit.,
p. 24, refers to
DE MORGAN
in the
Companion
to the
Almanac, which we have not seen.
2
Educational Times, 1868. On the Unsuitableness of Euclid as a Textbook of Geometry, London. * SYLVESTER'S Presidential Address to the Math, and Phys. Section of s
Laws of Verse, the Brit. Ass. at Exeter, 1869, given in SYLVESTER'S is a powerful answer to address This 120. eloquent 1870, p. London,
which knows
that
study Huxley's allegation that "Mathematics of induction, nothing of observation, nothing of experiment, nothing from Sylves sentences the We following causation." of quote nothing as to maintain that the habit of ter "I, of course, am not so absurd cultivated observation of external nature will be best or in any degree is
:
not
of the
ideas of
great all, by the study of mathematics." "Most, " Lagrange, modern mathematics have had their origin in observation." em has be expressed could quoted, than whom no greater authority if
the the importance to the mathematician of phatically his belief in a science of the eye mathematics called Gauss observation faculty of a thesis to show ever to be lamented Eiemann has written ;
;
the
and our that the basis of our conception of space is purely empirical, kinds of other that of result observation, the knowledge of its laws to exist subject to laws different from those space might be conceived are immersed." actual the space in which we which govern
A HISTORY OF MATHEMATICS
284
Thomas Simpson's geometry.
of
One nappy
event, however,
in 1870, of grew out of these discussions, the organization, the Association for the Improvement of Geometrical Teaching
G. T.). T. A. Hirst, its first president, expressed him self as follows: "I may say further, that I know no suc cessful teacher who will not admit that his success is (A.
I.
almost in proportion to the liberty he gives himself to and to give depart from the strict line of Euclid's Elements a life which, without that departure, it could the subject
not possess. I know no geometer who has read Euclid critito modes of exposi catty, no teacher who has paid attention are full of Elements Euclid's that tion, who does not admit defects.
They
*
swarm with
faults' in fact, as
was said by
the eminent professor of this college (De Morgan), who has helped to train, perhaps, some of the most vigorous thinkers of our time."
After
much
1
discussion the Association published in 1875 a
SyUabus of Plane Geometry, corresponding to Books I.VL of " Euclid. It was the aim to preserve the spirit and essentials of style of Euclid's Elements; and, while sacrificing nothing in rigour either of substance or form, to supply acknowledged The sequence of deficiences and remedy many minor defects.
that of Euclid, propositions, while it differs considerably from does so chiefly by bringing the propositions closer to the fun damental axioms on which they are based and thus it does ;
with Euclid's sequence in the sense of proving any theorem by means of one which follows it in Euclid's
not
conflict
order, though in many cases it simplifies Euclid's proofs by using a few theorems which are contained in the sequence of 2 The the Syllabus, but are not explicitly given by Euclid." mathethe of best the careful consideration received Syllabus 1
Mrst General Report, A.
3 Thirteenth
I.
G. T., 1871, p. 9.
General Meport> 1887, pp. 2223.
ELBMENTAEY GEOMETRY
285
matical minds in England; the British Association for the Advancement of Science appointed a committee to examine
the Syllabus and to cooperate with the A. I. Gr. T. In their report, in 1877, which favoured the Syllabus, was expressed the conviction that "no textbook that has yet been produced is
One
to succeed Euclid in the position of authority."
fit
of
the great drawbacks to reform was the fact that the great universities of Oxford and Cambridge in their examinations
upon a rigid adherence to the proofs and the sequence of propositions as given by Euclid thus no freedom was given to teachers to deviate from Euclid in any for admission insisted
;
" way. To be noted, moreover, was the total absence of origi " " or riders," or any question designed to determine the nals real knowledge of the pupil. But in the last few years the
work of the Association has been recognized by the universi It was noted above ties, and some freedom has been granted. that J. J. Sylvester was an enthusiastic supporter of reform. The difference in attitude on this question between the two fore most British mathematicians, J. J. Sylvester, the algebraist, and Arthur Gayley, the algebraist and geometer, was grotesque. a deeper than e'er plummet Sylvester wished to bury Euclid sounded" out of the schoolboy's reach; Cayley, an ardent admirer of Euclid, desired the retenMon of Simson's EudicL When reminded that this treatise was a mixture of Euclid and Simson, Cayley suggested striking out Simson's addifeMS and
keeping strictly to the original
The most
treatise,
1
difficult task in preparing the
SyUabusw&s the
treatment of proportion, Euclid's Book Y. The great admira tion for this Book Y. has been inconsistent with the practice
Says Nixon, "the present custom of though quietly assuming such of its results
in the schoolroom. omitting 1
Book
Fifteenth,
Y.,
General Report, A.
teenth General
eport, p. 28.
L
G. T., 1889,
p. 21.
See also
A HISTORY OF MATHEMATICS
286
1
Book VL, is " The testimony
as are needed in offers similar
book of Euclid
fifth,
:
invariably been
f
all
.
.
has
.
but the cleverest school
skipped/ by Here we have the extraordinary
2
boys."
Hirst
singularly illogical."
spectacle of pro
"
portion relating to magnitudes skipped," by consent of teachers who were shocked at Legendre, because he refers the
student to arithmetic for his theory of proportion this practice of English teachers a tacit
1
Is not
acknowledgment of
the truth of Hammer's 3 assertion that as an elementary text book the Elements should be rejected ? Euclid himself proba/
bly never intended them for the use of beginners. But the English are not prepared to follow the example of other nations and cast Euclid aside. evolution, not simplify,
The
and
first
by
revolution.
The The
British
mind advances by
British idea is to revise,
enrich the text of Euclid.
important attempt to revise and simplify the
fifth
book was made by Augustus De Morgan. He published, in " 1836, The Connexion of Number and Magnitude an attempt to explain the fifth book of Euclid." In 1837 it appeared as :
an appendix to his "Elements
of Trigonometry."
this revision that the substitute for
Syllabus, is modelled.
On
Book
It is
on
V., given in the
the treatment of proportion there
was great diversity of opinion among the members of the sub committee appointed by the Association to prepare this part of the Syllabus, and no unanimous agreement was reached. 4
There were, in the first place, different schemes for rearrang ing and simplifying Euclid's Book Y. According to Euclid four magnitudes,
a, b, c, d,
are in proportion,
1
when
for
any integers
E. C. J. NIXON, Eudid Bevised, Oxford, 1886, p. 9. fourth General Beport, A. I. G. T., 1874, p. 18. 8 Geschickte der Padagogik^ Vol. HI. see also K. A, SCHMID, Encyklopadie des gesammten Erziehungs und Uftierrichtswesen&, art. " Geometries 2
;
*
NIXON,
op. Git., p. 3.
ELEMENTARY GEOJIBTBT
m
and
we
n,
liave simultaneously ma=^=nb,
287
A
and me =nd.
definition of proportion adopted by the Italians Sannia and D'Ovidio was considered, according to which parts are sub
stituted for multiples, viz.
any integer m, a contains of times than c contains
5, c, d,
are in proportion,
if,
for
m neither a greater nor a less number
m
Another method discussed was
that of approaching incommensurables by the theory of limits, as adopted in the French work of E. Eouche and C. de Com
and by many American writers. This method, as also Sannia and D' Ovidio's definition of proportion, rests on the law of continuity which Clifford defines roughly as meaning " that
berousse,
all quantities
can be divided into any number of equal parts." *
If in a definition it
is
desirable to assume as little as possible,
Euclid's definition of proportion is preferable, inasmuch as
it
does not postulate this law. To minds capable of grasping Euclid's theory of proportion, that theory excels in beauty
the treatment of incommensurables with aid of the theory of limits. Says Hankel, "We cannot suppress the remark that the
now
prevalent treatment of irrational magnitudes in not well adapted to the subject, in as much as it geometry in the most unnatural manner things which belong separates is
to which, from its very together, and forces the continuous into the shacklec nature, the geometric structure belongs of the discrete, which nevertheless
asunder."
it
every
moment
pulls
2
About the time of Hankel a new logical difficulty came to It became evident that the theory of limits as it ex light. isted in the first half of the nineteenth century must remain unsatisfactory unless a theory of irrational numbers could 1
2
The Common Sense of the Exact
HANSEL,
op. c#., p. 66.
Sciences,
New York,
1891, p. 107.
A
288
HISTORY OF MATHEMATICS
As it was, the lines of procedure involved reasoning in a circle. The theory of limits was invoked to establish a theory of irrational numbers, whereupon the theory be presupposed.
of irrational numbers seemed to be used for a more complete exposition of the theory of limits. To avoid this vicious cir cle,
altogether
new
theories of irrational
numbers were devel
R
Dedekind, which oped by Karl Weierstrass, GL Cantor, and do not prerequire the theory of limits. 1 These developments
have greatly strengthened the logical foundations of mathe matics, but have not yet been put in a form suitable for use in elementary instruction. From the standpoint of the elementary teacher, a logical exposition of irrational numbers and of limits, appears impossible. Neither Euclid's treatment of propor tion, nor the treatment of irrational number and limits as given by modern logicians,
The quest
is
proper food for school children.
for perfect logical rigor is long
and disheartening.
doubtful whether mathematicians have yet reached absolute rigor. The creations of G. Cantor and Dedekind have
Indeed
it is
only recently given rise to contradictory results, to antinomies, that are thoroughly disquieting. To be sure, the introduction of certain restrictions in the definitions and the adoption of " have removed the antinomies advanced certain " axioms
thus far.
But we know not the day nor the hour when new In view of the subtlety of light. numbers and of doubt as to their real
antinomies will spring to the
modern
theories of
logical perfection, there
remains no alternative for the ele
mentary teacher but to reconcile himself with a logically imperfect treatment of the theory of limits and of irrational 1 Consult GEORG CANTOR, Contributions to the Founding of the Theory of Transfinite Numbers, translated, and provided with an Introduction and Notes, by Philip E. B. Jburdain, Chicago and London, 1915. Jour
dain's Introduction traces the history of the subject. Consult also GEORGE BRUCE HALSTED, On the Foundation of Technic of Arithmetic, Chicago,
ELEMENTARY GEOMETRY
289
1 In fact it is generally felt tliat th.es e subjects do numbers. not really belong to elementary mathematics, that a rudimen tary and purely intuitive treatment of limits is all that should
be attempted in high school and in the mathematics.
The same
first
conclusion on matters of logic
stages of college
is
reached from a
under the leadership of Giuseppe Peano of the University of Turin set themselves the problem to adopt definitions, axioms, and undefined terms different angle.
Italian mathematicians
so that geometry might be built
As
tion at
up without appealing
to intui
well known, Euclid quietly resorts to any point. intuition in many places, for the knowledge of certain facts.
In the very
first
is
proposition of his Elements he states without
proof that the two circles which he draws intersect each, other. As contributions to the logical exposition of geometry, the work of the Italian school was praiseworthy. movement is found in a treatise prepared representative of logical abstract
A
culmination of this
by the great German
thought in mathematics,
David Hilbert of the University of Gottingen. At the unveil ing of the GaussWeber monument in Gottingen, in June, 1899, Hilbert contributed a memorial address on the foundations of Geometry, entitled (hwndlagen der Geometrie, which was widely recognized as a masterpiece of logical exposition. The first edition was translated into English in 1902 by E. J. Townsend of the University of Illinois. Several German editions, some what enlarged, have appeared since. Hilbert sets up five groups of axioms, which include altogether twenty axioms. To show how Hilbert's axioms can serve as a foundation of geometry, built up without appeal to intuition, George Brace Halsted published in 1904 his Rational Geometry. This
bk
affords perhaps the easiest approach to a 1
cle,
knowledge of Hilbedfs
For an ffltiminatiag article on limits, consult BB&TRASTD RU&SBI/I,*& arti Infinity," in ffind, Vol. 33, London, 1908, p. 28&
"Mr. Ealteie on
A
290 system.
HISTORY OF MATHEMATICS
At the same time
it
demonstrates the utter impossi
bility of introducing high school students into geometry by such a route. To meet certain logical criticisms of his text,
Halsted prepared a thoroughly revised second edition in 190T, which has been translated into French.
RECENT MOVEMENTS IN TEACHING Toward the close of the nineteenth century the influence of the Association for the Advancement of Geometric Teaching " (A. I. G. T.), whose name was changed in 1897 to The Mathe matical Association " and whose work had been broadened in
scope, so as to include elementary mathematical instruction in
showed itself in many ways. To note the progress in geometry we need only examine such editions and revisions of Euclid as those of Casey, Nixon, Mackay, Langley and
general,
Phillips, Taylor,
and the Elements of Plane Geometry issued by
the Association. 1
At
the beginning of the twentieth century there were two great movements in the teaching of elementary mathematics. the other is a co One has been called the "
Perry Movement,"
operative movement of teachers of different nations under the name of the " International Commission on the Teaching of
Of these movements the first made at once a and lasting impress, especially in England and America, deep then and spent itself; the second was planned on a large scale and was still in the preliminary stages when interrupted by the great war which broke out in 1914. Mathematics."
1
For the history of geometrical teaching in the United States before and History of Mathematics in the U. S.,
1888, see CAJORI'S The Teaching 1880.
291
ELEMENTAKY GEOMETEY The Perry Movement
The reform which
this
movement introduced
is
the culmina
tion of thirty years of discussion in England, carried on by the members of the Mathematical Association. This culmination
centers around the personality of John Perry, then Professor of mechanics and mathematics of the Royal College of Science,
Perry was member of a large and strong committee of the British Association for the Advancement of Science,
London.
" appointed in 1901 to report on the Teaching of Mathematics." This was not the first time that the British Association had interested itself in this matter.
ported on "the
possibility of
A
similar committee
improving
had re
the methods of
instruction in elementary geometry" in 1873 and again in 1876. 1 These early reports considered the advisability of adopting an authorized syllabus of geometry; the second
report considered the merits of the syllabus prepared by the G. T. and recommended it ; nevertheless that syllabus
A. A.
had not been generally adopted at the beginning century.
Teaching
of the
continued to be guided mainly
by
new the*
requirements of examinations. It was at the Glasgow meeting of the British Association in 1901, when Section A (Mathematics and Physics) and Sec tion L (Education) held a joint meeting, that John Perry delivered a great address, which was followed by a brilliant discussion, leading to epochmaking results. An account of these deliberations was published in 1901 under the title,
Discussion on the Teaching of MatJiematics, edited by John Perry. This debate is far more spicy, interesting and valuable than
the formal report of the committee, made a year later, Perry called for a complete divorce from Euclid, for greater emphasis 1
See Reports of the British Association for the year 1873, page 459.
and for 1876, page
8.
A HISTOKY OF MATHEMATICS
292
for practical mensuration, foi the use of squared paper, for more solid geometry, for greatei emphasis upon the utilitarian parts of geometry. As a result, Euclid was dethroned in England from its position of complete
upon experimental geometry,
dominance as a textbook in elementary
schools.
In closing
the discussion, Perry said " I take it that we are all :
agreed upon the following points "1. Experimental methods in mensuration and geometry ought to precede demonstrative geometry, but even in the ear liest stages
" 2.
:
some deductive reasoning ought
to be introduced.
The experimental methods adopted may
to the
of the teacher
greatly be left include all those
judgment they may mentioned in the Elementary Syllabus which I presented. " 3. Decimals ought to be used in arithmetic from the begin :
.
.
.
ning.
"4.
The numerical
evaluation of
complex mathematical
expressions may be taken up almost as part of arithmetic or at the beginning of the study of algebra, as it is useful in familiarizing boys with the meaning of mathematical symbols. "5, Logarithms may be used in numerical calculation as soon as a knows a n x am am + n and before he is able
boy
=
,
long
to calculate logarithms. But a boy ought to have a clear notion of what is meant by the logarithm of a number. " 6.
In mathematical teaching a thoughtful teacher may be
encouraged to distinguish what is essential for education in the sequence which he employs, from what is merely accord ing to arbitrary fashion, and to endeavor to find out what sequence is best, educationally, for the particular kind of boy
whom he
has to teach.
"7. Examination cannot be done away with in England. Great thoughtfulness and experience are necessary qualifica tions for an external examiner. It ought to be understood that an examination of a good teacher's pupils by any otfeter
ELEMENTARY GEOMETRY examiner than the teacher himself tion.
.
of
A
am
in
more doubt.
thoughtful teacher ought to
squared paper and easy algebra,
dynamics and laboratory experiments, to
an imperfect examina
.
.
" About these that follow I "8.
is
293
know by
.
that
.
.
by the use
illustrations
it is
from
possible to give
young boys the notions underlying the methods of the
Infinitesimal calculus.
A
thoughtful teacher may freely use the ideas and symbolism of the calculus in teaching elementary mechanics
"9.
to students.
A
thoughtful teacher may allow boys to begin the formal study of the calculus before he has taken up advanced
"10.
algebra or advanced trigonometry, or the formal study of analytical or geometrical conies, and ought to be encouraged to use in this study, not merely geometrical illustrations, but
from mechanics and physics, and illustrations from any other quantitative study in which a boy may be
illustrations
engaged." In the United States the Perry movement did not bring about as many changes as in England. There are two prin cipal reasons for this.
One reason
is
that Euclid had been
discarded long ago as a text for beginners, hence the agitation in Great Britain on that point had no bearing on American second reason is that some of the otfeer recom conditions.
A
mendations of Perry did not directly fit the American sitaatloB. For example, Perry s warfare against the system of examina tions then in vogue in British universities and also in the ?
English tions
The
civil service
did not affect America, where examina differently and received less stress.
were conducted clearest
American declaration with respect to the Perry came
in so far as it affected the United States,
movement, from E. EL Moore
of the University of Chicago in his presi
A HISTORY OF MATHEMATICS
294
dential address,
livered
"On
on December
matical Society. 1 matics,
when he
the Foundations of Mathematics," de 29, 1902, before the American Mathe
Moore had in mind mainly college mathe "In agreement with Perry, it would
said:
seem possible that the student might be brought into relation with the
fundamental
elements
vital
of
trigonometry, analytic geometry, and the calculus, on condition that the whole treatment in its origin is and in its development remains closely
associated
with thoroughly
concrete
phenomena."
With reference to secondary schools, Moore puts the question " Would it not be possible to organize the algebra, geometry,
:
and physics of the secondary school
into a thoroughly coherent four years' course, comparable in strength and closeness of structure with the four years' course in Latin?" As to
method of instruction
:
" This program of reform
calls for
the
development of a thoroughgoing laboratory system of instruc tion in mathematics and physics, a principal purpose being as far as possible to develop on the part of every student the true spirit of research, and an appreciation, practical as well as theoretic, of the fundamental methods of science."
Experience seems to show that the success of the laboratory method of mathematical instruction rests upon several contin gencies, such as, (1) rearrangement of the schedule so as to allow more time in school for the courses in mathematics, (2) the
equipment of suitable
of teachers
laboratories, (3) the enlistment
who
are competent to give laboratory courses, and so on. Even if all these conditions prevail, there is no a priori reason why the laboratory method should excel other methods.
The
superiority of
any method over its competitors must be Such superiority has not yet been
determined by actual trial
established experimentally for the laboratory method. 1
This address was printed in Bulletin of the Am. Math. Society, Yol. 9, 1903, pp. 402424, also in Science, N. S. t Vol. 17, 1903, pp. 401416.
ELEMENTARY GEOMETRY
295
We
pass to the broader question, the unification of the dif ferent branches of high school mathematics, carried on in a manner that will meet more effectively the practical demands
This question has been under discussion and ex perimentation ever since the opening of the new century. The of the age.
impression seems to prevail that, in past decades and past cen mathematics never were unified, that teachers placed
turies,
mathematical topics into separate and airtight compartments. truth. Previous to the nine
Nothing can be further from the
teenth century there existed a fair degree of fusion of different branches of mathematics. We see this, for instance, in the
Annies papyrus, in Euclid's Elements, with its three books on arithmetic, in the works of Yieta, Stevin, Oughtred, Newton, Leibniz, and Euler.
The geometric phraseology "linear"
equation, "quadratic" " for a 2 " a cube " for " cubic " " a , equation, equation, square a3, shows that, in its formative period, algebra was not disso ciated from geometry. Moreover, the solution of cubic and
quartic equations by the method of geometric construction ap pears earlier in the history of mathematics than do the purely algebraic solutions. Isolation in mathematics as a science is a product of the nineteenth century. Steiner and Yon Staudt isolated modern geometry from analysis. On the other hand,
Weierstrass, Kroneeker, Klein, and others worked in the direc tion of arithmetization
which makes number the
tion of mathematics.
This isolation
sole founda due to the efforts toward increased rigor of presentation, in course of which fee foundations of mathematics were examined anew. Not only is
in the growth of mathematics as a science, bit also in the teaching of mathematics, isolation of topics set in. frequently, books which in their first editions exhibited a certain degree of
fusion of topics, passed to greater isolation in later editions. Thus, Legendre's Geometry, first published in 1794, stood at
A HISTORY OF MATHEMATICS
296 first for
a moderate unification of geometry with, arithmetic As late as 1845 this text still contained
and trigonometry.
trigonometry, but later the trigonometry and the practical applications of geometry were gradually filtered out. Nicholas
New and
Complete System of Arithmetic, printed at Newburyport, Mass., in 1788, and used for a time even in our American colleges, contained in addition to the ordinary arith
Pike's
metical topics, a treatment of logarithms, trigonometry, alge and conic sections. The fourth edition of Pike's book
bra,
omitted the topics not strictly arithmetical, and constituted Pike's Abridged Arithmetic. Thus there arose in teaching an isolation of topics, a moderate degree of which was beneficial.
Teachers found
it easier to reach good results in the classroom one by taking topic at a time. But isolation came to be car ried to an extreme. The exercises of the schoolroom related
The problems of practical life called for combine arithmetic, algebra, and geometry. The in lacked and skill combination. Hence the pupils experience need for reunification. 'No agreement has yet been reached as to isolated topics.
ability to
to the kind
and degree of
unification that is
most suitable
under modern conditions of industry and life. There is dan ger of excessive unification. Too much fusion may produce confusion. In intense fusion there is danger of introducing simultaneously a large number of new ideas, new definitions. The pupil does not concentrate upon a few ideas, but scatters his efforts over a large superficial
number of them and
obtains only a
view of any one.
If it be asked what degree of unification can be safely adopted as having stood the test of experience, the answer would seem to be as follows : At any stage of mathematical study, let the
pupil use all the mathematics that he has
For instance, when he studies algebra,
had up
let
to that time.
him use
freely the
arithmetic that preceded, as well as the intuitive geometiy
ELEMENTARY GEOMETRY
297
Mm
use previously learned when pursuing solid geometry, let the arithmetic, algebra, and plane geometry previously studied. ;
To this extent, unification is sanctioned by experience. much farther we can go advantageously under modern tions
remains to be determined by experience.
How condi
Doubtless
some allowance must be made for individual predilections teachers and the practical needs of different pupils,
of
The International Commission
At that
the Fourth International Congress of mathematicians at Eome in 1908, David Eugene Smith of Teach
was held
ers College, Columbia University, made a motion calling for the appointment of an International Commission on the Teach ing of Mathematics. The motion was carried and a Central
Committee was appointed, consisting of Felix Klein of the University of Gottingen, Sir George Greenhill of London, and Henri Fehr of the University of Geneva in Switzerland. In
1912 David Eugene Smith was added to the committee. The Commission adopted as its official organ IS Enseignement mathematique, Mevue international^ paraissant tons
les
deux mois.
bimonthly journal had been started at Paris in 1899 pears under the editorship of C. A. Laisant and H. Fehr. far the
Commission has
j
This it
ap Thus
chiefly occupied itself in effecting
organizations in the different countries, and in collecting and publishing, for comparative examination, the fullest possible
information as to the present status of the teaching of mathe matics in each of the countries represented. The first
report
of the International Commission was
made
at the Fifth Inter
national Congress of Mathematicians, 1 held at Cambridge, Eng At that time there were 27 countries repre land, in 1912. sented, 1
and 150 printed reports were made, leaving about 50
3?or an account of fee work of this congress, see J. W. A. YOTTHG in School Science and Mathematics, Vol. 12, 1912, pp. 702715.
A HISTORY OF MATHEMATICS
298 still
to
be published.
These reports contain detailed informa
on the condition of mathematical teaching at different schools and in different countries. They are important contri tion
butions to the history of education. Most exhaustive are the numerous reports prepared by the Germans. The British re ports are models of concise presentation.
The documents pre
pared in the United States by special committees acting under the direction of the American Commissioners (D. E. Smith of Teachers College, W. F: Osgood of Harvard University, J. W. A. Young of the University of Chicago) have been published by the United States Bureau of Education. The breaking out of the Great
War,
in 1914, interrupted the
work
of the Inter
national Commission.
While the publications 1 hitherto issued are mainly historical, and descriptive of present conditions, it is evident that the representatives of the international movement stand for the more thorough embodiment, in mathematical teaching, of cer tain ideas, at least a part of which were emphasized in the Perry movement. Predominant are the ideas of graphic repre sentation, functionality, conservative unification of mathe matics, alignment with the practical demands of the age. In these ideas were strongly advocated by F. Klein at a school conference held in 1900. In France these same ideas
Germany
found expression in the textbooks published by E. Borel in 1903, which were translated into German by P. Stackel in 1 Hie French reports are entitled, Eapports de la Commission Interna tionale de rSnseignement Mathematique, Hachette & Company, Paris. The English reports are brought out under the heading The Teaching of
Mathematics in the United Kingdom, being a series of papers prepared for the International Commission on the teaching of mathematics, Wyman & Sons, Fetter Lane, London, E. C. The German reports are Abhandlungen uber den mathematischen Tfnterricht in Deutschland, Veranlasst durch die Internationale mathematische TTnterrichtskommission. Herausgegeben von F. KLEIN, Vols. IV, B. G. Teubner, Leipzic u. Berlin.
ELEMENTARY GEOMETRY 1908, to serye as illustrations of
of actual realization of the
The
299
what can be done in the way
new
plans. subject of graphic representation of mathematical ex
pressions involving variables goes back to the time of Kewton the graphic representation of curves defined by certain properties goes back to the time of the Greeks.
and Descartes
;
In the nineteenth century graphic methods came to be used 1 widely in mechanical science, not only abroad but also in the United States. In 1898 R. H. Thurston of Cornell University read before the American Society of Mechanical Engineers a pappr on "Graphic Diagrams and Glyptic Models." In the teaching of elementary mathematics graphic methods were introduced slowly. Lone voices advocating the use of graphs in teaching were heard in the nineteenth century. In 1842 Strehlke of Danzig expressed himself as follows: 2
"The
graphic representation of functions appears to
me an
effective aid in instruction, for the purpose of conveying
a
With this end in view the number of such graphs
clear conception of their properties.
I have drawn a considerable
curves of sines, cosines, tangents, of natural and Briggian x2 x, y logarithms. . . . Even simple functions like y y a?, y #*, if they are drawn for the same variable and
=
,
=
=
upon the same not see the
which the pupil would put before him." In the
sheet, present properties
moment
the function
is
solution of equations, graphic methods received recognition in L. SaintLoup's Traitd de la resolution des &qwatiofis num&riques, Paris, 1861, a text written for students in the Polytechnic "a School and the Normal School in Paris, Fusage des eanditats
aux
ecoles polytechnique et
norm ale." The students were
See the " Second Report on the Development of Graphic Methods in 1 Mechanical Science,' by H. S. HBLE SHAW, in Beport of the British Association for the Advancement of Science for the year 1892, pp. 373531. 1
2
Grunert's Archiv, Vol. 2, 1842, p. 111.
A HISTORY OF MATHEMATICS
300
more advanced than are American students grade. (r.
of high, school
In the Tutorial Trigonometry, by William Briggs and
H. Bryan, London, 1897, attention
is
given to the tracing
of the curves of the trigonometric functions.
In 1896 appeared Arthur H. Barker's Graphical Calculus, London, which represented a series of lectures given to en
marks a complete contrast to the ordinary mathematical book of the school of Todhunter. It is a work intended for students ready to enter upon the study gineering students.
It
of the calculus.
In the United States the
earliest use of graphic
methods in
elementary teaching consisted in the representation, by curves, of empirical data, such as temperatures, cloudiness of the sky.
From 1885
methods were taught in the school of the Chautauqua Town and Country Club, to about two thou sand pupils. 1 But these graphic methods met with no general to 1889 such
adoption at that time.
In 1898 Francis E.
Mpher
of
Wash
ington University in St. Louis brought out an Introduction to Graphic Algebra. The first regular textbook using graphs was (1899) by George W. Evans of Boston. In 1902 there appeared as a supplement to Webster Wells' Essentials of Algebra and New Higher Algebra, a monograph
the
Mgebrafor Schools
of twentyone pages on Graphs, prepared by Eobert J. Aley, then of the University of Indiana. Henry B. Newson of
the University of Kansas followed in 1905 with a Graphic In Vol. 5 (1905) of School Algebra for Secondary Schools.
and Mathematics there is an article on graphs by Touton of Kansas City. Since 1905 our algebras for secondary schools introduce the graph as a regular subject for
Science F. C.
instruction.
As
" early as 1903 the use of the graphic methods
" See CHARLES BARNARD, Graphic Methods in Teaching," Educa Monographs published by the N. T. College for the Training of Teachers, Vol. II., No. 6, 1889, pp. 209223. 1
tional
ELEMENTARY GEOMETRY
301
and is
illustrations, particularly with the solution of equations, also expected " as an entrance requirement in mathematics
to eastern colleges.
American Associations There have been organized in the United States a number of associations for the advancement of the teaching of mathe matics and science in secondary schools. Thus we have the Association of Teachers of Mathematics in New England, the Association of Ohio Teachers of Mathematics and Science, the Association of Teachers of Mathematics in the Middle States and Maryland, Central Association of Science and Mathematics Teachers, Colorado Mathematics Association, Kansas Associa tion of Mathematics, Missouri Society of Teachers of Mathe matics and Science, Iowa Association of Mathematics Teachers,
North
Dakota
Association
of
Science
and
Mathematics
Teachers, Southern California Science and Mathematics Asso ciation, Oregon State Science and Mathematical Teachers Association.
The Mathematical Association
of America,
organized in
mainly to the teaching of Math The American Mathematical Society (or
December, 1914, devotes ematics in College. ganized in 1888 as the reorganized under
its
itself
New York present
Mathematical Society, and
name
in 1894) devotes itself
entirely to the furtherance of original researdh. The National Educational Association (organized in
delphia in 1857 under the
name
PMIa
of " National Teachers* Asso
" and renamed in 1870) has always interested itself in the teaching of elementary mathematics. In 1893 a "Com ciation
mittee of Ten" reported on secondary school studies, including the ^commendations of various subcommittees or "confer " conference of ten " on mathematics met at Har ences."
A
vard in December, 1892.
Simon Newcomb acted as chairman.
A HISTORY OF MATHEMATICS
302
A
report was prepared, which
is
given in outline in the Report
of the Commissioner of Education, 18921893, Vol. 2, p. 1426. Pull reports of the " conferences " were published separately in 1894. It was recommended by the mathematics conference that the course in arithmetic be abridged
by omitting
subjects
which perplex and exhaust the pupil without affording mental discipline, and enriched by exercises in concrete problems, that geometry be first taught in concrete form, including measure ments, and that greater care be taken to avoid slovenliness of expression in demonstrations. Stress was laid upon the inter
and geometry. The eagerness with which the various reports were received caused a " Com mittee of Fifteen" to be appointed by the National Educa
lacing of arithmetic, algebra,
Association to institute similar inquiries for schools below the high school. Eecommendations were made on the
tion
1 It was argued that the simplest frac teaching of arithmetic. could be tions, ^, ^, ^, taught very early, that compound in terest, foreign exchange, and certain other topics be dropped,
that mental arithmetic be emphasized, that there be a re arrangement of subjects, and that the rudiments of algebra be
introduced in the seventh year. In 1899 the Committee on College Entrance Requirements, appointed by the N. E. A., made recommendations in mathe matics which were in practical agreement with those of the Committee of Ten. In 1904 the Association of Mathematics
Teachers in
W. Evans
New England appointed
a committee, with George
of Boston as chairman, to consider the advisability
some syllabus already drawn up. Such a syllabus was drawn up; though carefully prepared, it did not become generally known. A of preparing a geometry syllabus, or adopting
by a syllabus committee of the Central Association of Science and Mathematics Teachers, made in 1906, attracted
report
1
Report of the U.
S.
Commissioner of Education for 18931894,
22LEMENTARY GBOMETBY
303
was imperfect in some respects. Finally and 1909 a committee was organized under the joint auspices of the National Education Association and the American Federation of Teachers of the Mathematical and wider attention, but
in 1908
H. E. Slaught of the University of Chi The final report of this committee was printed in 1912, and received much attention both in the United States and abroad. Theorems are classified into four National Sciences.
cago acted as chairman.
groups, according to their importance, the groups being dis tinguished one from the other by the kind of type used. The
report gives considerable attention also to logical considera tions, and contains an introduction on the history of the teach ing of geometry during the last two centuries.
In recent time there has been considerable unrest in the United States on matters relating to mathematics in second ary schools. Proceeding on the theory that the real aim is " train the mind," but to teach only not, as formerly held, to
"
things that are practical," so that the mathematical instruc tion can be " hitched up to life," some educators exert pressure to omit certain subjects formerly thought essential, such as
the treatment of factoring and fractions in algebra, and to restrict problems in algebra and exercises in geometry to those that are practical. In some ways the modern tendency re sembles the movement of the fifteenth and sixteenth centuries, led
" by the Commercial School
of Arithmeticians." " is meant
sixteenth century, so now, there
that which
is ultimately practical,
by
but that which
As
in the
practical," not is
immediately
a course resembling ancient Egyptian By practical. geometry makes a stronger appeal than one resembling the more scientific Greek geometry. There are other modern this test
who go farther and, denying the disciplinary value of mathematics, insist that high school mathematics, or at least of it, should be removed from the required list of high a educators
part
A HISTORY OF MATHEMATICS
304 school studies.
It
may
be of interest, in this place, to call
attention to earlier attacks on the value of mathematics as an exercise in the training of the mind.
Attacks upon the Study of Mathematics as a Training of the
Mind. 1
Probably the most famous attack that has ever been made upon the educational value of mathematics was published by the philosopher, Sir William Hamilton, in the Edinburgh Re view of 1836. He claims that mathematics is "not an im " " If we consult reason, experience and the proving study ;
common testimony
intellectual studies tend to J
and modern times, none of our cultivate a smaller number of the
of ancient
a more partial manner, than mathematics." He " proceeds to adduce testimony to the effect that the cultiva tion afforded by the mathematics is, in the highest degree, onesided and that mathematics " freeze and faculties, in
parch
contracted,"
the mind," that this science is "absolutely pernicious as a mean of internal culture." Hamilton's argument proceeds
along two principal lines the first is to prove that those who confine their studies to mathematics alone are addicted to blind ;
credulity or irrational scepticism and, in general, lack good admit at once, that this last judgment in affairs of life.
We
the exclusive study of any branch of knowledge is to be discouraged as undesirable for a liberal education. Hamilton's second mode of attack consists in try
contention
is
probably valid
ing to show that
;
many mathematicians found mathematics
unsatisfactory as an exercise of the mind, and renounced it. The only mathematicians of sufficient prominence to be known 1 See F. CAJORI, " A review of three famous attacks upon the study of mathematics as a training of the mind," in the Popular Science Monthly, 1912, pp. 360372. See also Dr. C. J. KETSER, Mathematics, Columbia
Univ. Press, 1907, pp. 2044.
ELEMENTARY GEOMETRY
305
modern reader are D'Alembert, Pascal, Descartes, and Dugald Stewart. Many other men, from all ages and nations, " are quoted, and constitute indeed a cloud of "witnesses." Says " I am ashamed to confess that Alfred Pringsheim of Munich to the
:
before reading Hamilton's article I did not know a single one of these great authorities even by name an extenuating cir ;
cumstance
is
the fact that some of these names I could not
find even in the scientific directories."
The most extensive
examination of Hamilton's quotations and most thorough dis proof of his contentions are found in an article by A. T. Bledsoe in the Southern Review for July, 1877. Bledsoe was at one time professor of mathematics at the University of Virginia.
He
proves what some other writers before him hinted at, or proved only in part, namely, that Hamilton was extremely careless in the selection of his quotations. By making partial extracts, badly chosen,
he made
scientists
1 posite of their real sentiments. of his treatment of Descartes.
Most
say exactly the op
strikingly
is this
true
The second great attack upon mathematical study was made by Schopenhauer, the pessimistic sage of TrankfortontheMain. In his The World as Witt and Idea, second edition, 1844, he makes war on Euclid and Ms geometrical demonstra tions.
"
We
to admit that
are compelled by the principle of contradiction what Euclid demonateates is true, but we do not
comprehend why it is so. We have, fee^efore, almost Ae same uncomfortable feeling that we experience after ^ juggling trick, and, in fact, most of Euclid's demonstrations are re markably
like
such
feats.
The
truth almost always eatery
by
the back door, for it manifests itself par acddens l&rough. soine contingent circumstance. Often a reductio ad dbswrctem shuts all i
Sir
the doors one after another, until only one
is left>
which
On this point consult also JOHN STUABT Mitt's An Examination of WWiam Hamilton's Philosophy, New York, 1884, VoL EL, pp. 30633&
A HISTORY OP MATHEMATICS
306
we
This is part of Schopen famous characterization of mathematical reasoning as His objections "mousetrap proofs" (Mausefallenbeweise). are therefore compelled to enter."
hauer's
are directed almost
entirely against
He had
Euclid.
no
His criticism of acquaintance with modern mathematics. Euclid as a text for children is largely valid. Euclid did not write for children. plain,
how
It
is
a historical puzzle, difficult to ex
Euclid ever came to be regarded as a text suitable
for the first introduction into geometry.
As
to the reductio
ad
Be Morgan said: "The most absurdum method of proof. serious embarrassment in the purely reasoning part is the ad absurdum, or
reductio
though
it
indirect demonstration.
This form
generally the last to be clearly understood, occurs almost on the threshold of the elements."
of argument
is
Schopenhauer's attack bears only indirectly upon the question relating to the mindtraining value of mathematics. His views
must have attracted considerable attention in Germany,
for as
1894 A. Pringsheim thought it necessary to refute his argument, and only nine years ago Felix Klein referred to late as
him at some length
in a
mathematical lecture at the University
of Gottingen.
A
third attack
1869, p.
upon mathematics was made
in 1869
by the
Thomas H. Huxley.
In the Fortnightly Review, 433, he said: Mathematics is that study "which
naturalist,
knows nothing
of observation, nothing of experiment, nothing
of induction, nothing of causation," In an afterdinner speech, reported in MacmiUan's Magazine, 1869, pp. 177184, he argued
against language and mathematical training, and in favor of
"Mathematical training," he said, "is almost purely deductive. The mathematician starts with a few propositions, the proof of which is so obvious that they scientific
training.
are called selfevident, and the rest of his work consists of subtle deductions from them." Of great interest is the reply
ELEMENTARY GEOMETRY made
307
1 Huxley by the mathematician, J. J. Sylvester. To Americans Sylvester's name is memorable, because in 1876 Sylvester came over from England and for eight years taught modern higher algebra to American students at the Johns Hopkins University and gave a powerful stimulus to the study
to
of higher mathematics in this enthusiast.
His reply
before Section
own
A
to
country.
Huxley was
Sylvester was an
his presidential address
of the British Association in 1869.
experiences as a mathematician he tried to
By
his
show that
Huxley was wrong. analysis
is
Sylvester insists "that mathematical constantly invoking the aid of new principles, new
and new methods . springing direct from the inher ent powers and activity of the human mind that it is unceasingly calling forth the faculties of observation and com ideas
.
.
.
parison, that one of
its
.
.
principal weapons is induction, that
it
has frequent recourse to experimental trial and verification, and that it affords a boundless scope for the exercise of the highest efforts of imagination and invention. Lagrange expressed emphatically his belief in the importance to the .
.
.
mathematician of the faculty of observation Gauss called mathematics the science of the eye ... I could tell a story of almost romantic interest about my own latest researches in a ;
field
where geometry, algebra, and the theory of numbers melt manner into one another, . . which would
in a surprising
very
.
strikingly illustrate
how much
observation, divination,
induction, experimental trial, and verification, causation too, . have to do with the work of the mathematician.^ Syl . .
vester proved conclusively that the mathematician engaged in original research does exercise powers of internal observation, induction, experimentation,
and even causation.
The
vital
question arises, are these powers exercised by the pupil in the classroom? That depends. If a textbook in geometry is i
See page 283, last footnote.
A HISTORY OF MATHEMATICS
308
the case in some schools, then doubt But when attention is paid to original the use of measuring instruments, to modes of
memorized, as used to less
Huxley was
problems, to
"be
right.
proving theorems different from those in the book, then the pupil exercises the same faculties as does the advanced mathe matician engaged in research. Perhaps the latest prominent attacks on mathematics are by David Snedden, Commissioner of Education in Massachusetts, in
an address delivered in 1915 before the National Educa and by Abraham Flexner in an article,
tion Association,
"A
called
Modern
School."
1
Placing great faith in the
methods of experimental pedagogy that are advocated in schools of education of our universities at the present time, these writers are attacking the mindtraining value of mathe matics and are insisting that, in view of the results obtained
from experiments, the burden and geometry should remain rests
those
of proof to
show
that algebra
in the high school curriculum
upon those who advocate its continuance and not upon who would eliminate them. Says Flexner: "Modern
education will include nothing simply because tradition recom
mends
it,
...
it
includes nothing for which an affirmative
cannot now be made
out.
As has already been
intimated,
method of approach would probably result in greatly re ducing the time allowed to mathematics, and decidedly chang this
ing the form of what is still retained. If, for example, only so much arithmetic is taught as people actually have occasion to use, the subject will sink to modest proportions. The .
.
.
same policy may be employed in dealing with algebra and geometry. What is taught, when it is taught, and how it is taught will in that event depend altogether on what is needed, * of Reviews, VoL 53,
1916,
pp.
466474.
See
also
FLEXNBR'S "Parents and Schools," in the Atlantic Monthly, July, pp. $533.
A.
1916,
ELEMENTARY GEOMETRY when
it is
needed, and the form in which
will be instructive to
Certain
it
is
309
it is
watch the outcome of
needed."
It
this discussion.
that the teaching of mathematics will require
constant modifications in order to keep pace with the ever
changing needs of modern
life.
INl)EX
9395, 101103
Racists, 115, 118, 119, 188. Abacus, 11, 14, 17, 2628, 3741, 106, 112, 114, 117119, 186. Abel, 227, 238.
183, 206,
Absolute geometry, 273. See NonEuclidean geometry. Abu Dscha 'far Alchazin, 110. Abu Ja'kub Isfcafc ibn Hunain, 127.
Abu'l Dschud, 110. Abul Gud. See Abu'l Dschud. Abu'l Wafa, 104, 107, 128, 130, 248. Achilles
and
tortoise,
paradox
of, 59.
Acre, 178, 179. Aaalbold, 132. Adams, D., 218, 219, 222. Addition, 26, 37, 38, 96, 101, 105, 110, 115, 145. See Computation. Adelhard of Bath. See Athelard of
224245,
209,
283,
122,
156,
288;
Algoristic school, 118, 119, 188.
Algorithm. See Algorism. Al Hovarezmi. See AlchwarizmiAliojiot parts, 23, 197, 218. Alkalsadi, 110, 150.
Al Karchi,
106, 110.
Alligation, 110, 185, 198. Allman, 47, 48, 50, 51, 53, 56, 57, 59, 62, 63, 66.
Al Mahani, Al Mamun,
110. 126, 128.
Almansur,
129.
Alnager, 176. Alsted,203.
Bath.
Amicable numbers, Amyclas, 63.
29.
Analysis, in ancient gseojB0try,61,62; arithmetic, 196, 197; algebraic.
1925, 28, 34, 4345, 47, 82,
m
120,222. A. L G. T., 68, 69, 206, 208, 213, 263, 268, 279, 284286, 288.
Akhmim
in Arabia, 104, 106
origin of word, 107. See Notation. Algorism, 119; origin of word, 105.
Aeneas, 172. Agrimensores, 8992.
Ahmes,
;
Middle Ages, 118, 120, 133; Modern Times, 139, 141, 111;
249.
Anaxagoras, 48, 49, 53. AnaaimaBder, 48.
papyrus, 25, 82.
Albanna, ibn, 106, 111, 150, Al Battani, 118, 130. Albategnius. See Al Battanl.
Anaxiraenes, 48. Angle, triseetioa of, 34, $5 134, 135,
Albiruni, 15, 104. Alehwarizml, 104^109, 118, 128, 129,
Anharmonlc
226. ratio, 258* Annuities, 198. Anthology, Palatine, 33, 34, 113*
133,208. Alcuin, 112114, 119, 131, 220.
Antiparallel, 262.
Antiphon, 57459. Apian, 141. Apices of Boethius, 12, 14^16, 112, 114*
Aldschebr walmukabala, 107, 108. Alexandrian School, First, 6482; Second, 8289. Algebra, in Egypt, 2325; in Greece, 26, 3337;
115, 118.
Apollonian Problem, 79.
in Borne, 42; in India,
811
INDEX
312
Apollonius, 65, 75, 7880, 86, 87, 104, 127, 247, 250, 255.
Apothecaries' weight, 173. Appuleius, 32. Arabic notation. See Hindu notation.
Arabs,
10, 1318, 24, 65, 84, 95,
97,
103111, 116, 118, 119, 124132, 134, 137, 147, 150, 188, 224, 238.
Archimedes,
28, 33, 6365, 73, 7579, 86, 127, 134, 158, 242, 250, 254. Archytas, 53, 60, 62, 63.
Baillet,25.
Baker, 187, 193. Bakhshali arithmetic, Ball,
W. W.
94, 98, 99. R., 16, 133, 180, 184, 206
242,281. Baltzer, R., 273, 280. Bamberg arithmetic, 16, 140, 180.
Barleycorn, 175, 177.
Barreme,
192.
Barter, 198.
Arenarius, 29.
Base, logarithmic, 160. Based ow, 197.
Argand, 243.
Battaglini, 278.
Aristotle, 29, 47, 57, 61, 76, 132, 275. Arithmetic, in Egypt, 1926, 82; in Greece, 2637; in Rome, 3742; in
Bayle, 246. Bede, 112, 119. Beha Eddin, 128. Bekker, 57. Benese, R. de, 178. Berkeley, 144.
India, 93101; in Arabia, 103111; in Middle Ages, 111122 ; in Modern
Times, 139221, 145168 in England, Reforms in teach 20, 168, 179211 ing, 211219 in the United States, 215223. See Computation, Hindu ;
;
;
notation, tems.
Notation,
Numbersys
Arithmetical triangle, 238. Arneth, 125. Arnold, M., 207. Arnold, T., 207. Artificial numbers. See Logarithms. Aryabhatta, 12, 94, 99, 100, 102, 123. See False Assumption, tentative. position.
^Athelard of Bath, 104, 118, 132,133,
Bernelinus, 114, 115. Bernoulli, James, 237. Bernoulli, John, 200.
Bevan, B., 259. Beyer, 153. Bezout, 267, 276.
Bhaskara,
50, 95, 99, 101, 102,
Billingsley, 246, 281. Billion, 144. Binomial surds, 103.
Binomial theorem, 209, 229, 237, 238; not inscribed on Newton's tomb, 239.
Biot, 257.
136.
Attic signs, 7.
Biquadratic equations, 226. Bitonto, 267. Bockler, 164.
Austrian method, of subtraction, 213;
Boethius, 12, 14, 15,
Athenaeus, 63, Atkinson, 275.
of division, 214, 215.
Avoirdupois weight, 173, 174, 196, 198. Axioms, 46, 61, 67, 6971, 89, 90, 275, 276, 279*, 281. See Postulates, Par allelpostulate.
Babylonians,
1, 5, 6, 8, 9, 10,20, 23, 30, 43, 44, 49, 79, 84, 88, 123, 175.
Bache,215. Bachet de Meziriac, 220, 222.
Bachmann, 265. Backer rule of three, Bacon, R., 137.
103,
123,205.
198.
16, 32, 41, 91, 92,
111, 112, 114, 115, 117, 118, 131, 13&, 133, 136, 142, 147, 188.
Bolyai, J., 70, 71, 86, 127, 268, 269^
273,274,275,279. Bolyai, W., 273. Bombelli, 226, 227.
Boncompagni,
105, 142.
Bonnycastle, 244, 267.
Bookkeeping, 121, 168, 188, 206. Borelli, 268. Boscovich, 266. Bonelles.
See BouveUes.
Bourdon, 208.
INDEX
313
Bouvelles, 247.
122, 127, 129, 130, 132, 133, 134, 136,
Bovillus, 247.
137, 141145, 150, 181, 184, 194, 222,
Brackett, 208.
Bradwardine, 136, 137, Brahe, Tycho, 250. Brahmagupta, 94, 95,
247, 250.
227, 231, 232, 235, 238, 245, 247249, 250, 252, 282.
Capella, 91, 111, 131. 100, 101, 102,
129, 196, 205.
,
Cardan,
150, 151, 222, 224, 225, 227, 228, 229, 230, 232, 234, 240.
Bretschneider, 47, 50, 56, 62, 123.
Cardano.
Brewer, 172. Brewster, 239, 277, 283. Brianchon, 248, 252, 253, 257, 260. Brianchon point, 257. Brianchon 's theorem, 257.
Carnot, L., 83, 257, 369. Casey, 259, 288. Cassiodorius, 91, 92, 111, 112.
Bridge, 244. Bridges, 188. Briggian logarithms, 159, 162, 165, 199.
Castellucio, 146.
Casting out
tjie 9's, 98, 106, 149, 184,
185.
Cataldi, 150, 187.
Cattleproblem, 33.
Cauchy, 244.
Briggs, 161163, 167, 179, 238, Briot and Bouquet, 227.
Cavalieri, 209.
Brocard, H., 259, 261.
Broeard circle, 261, 262. Brocard points and angles, 261. Brocard' s first and second triangle, 261.
Caxton, 16, 180. Cayley, 285. Ceva, 83, 256. Ceva's theorem, 256. Chain, 179. Chainrule, 196, 197.
Brockhaus,
159.
Brocki, J.
See Broscius.
Brown,
See Cardan.
Champollion, 6. Charlemagne, 168, 220. Charles XIL, 2, 205.
200.
Broscius, 247. Brute, 172. Bryson of Heraclea, 58, 59.
Buckley, 150, 187, 188,
Buddha,
Chasles, 74, 83, 87, 89, 248, 252, 255, 256, 258, 259,
Chauvenet, 278. Chebichev, 31.
189.
Chinese. 94, 123, 240.
29.
Buee, 243. Burgess, 12.
Omquet,
Biirgi, 153, 166. Burkhardt, 226.
Cipher, 152; origin of word, 119.
deero,
CM% 48* 52, 53, 54, 57, m> 59, 73, 75, 78, 79, 87, 20, 255, 265, 270 ; division of, 10, 43, 73, 74, 7^84,12^ 134, 265.
Bushel, 172.
Butterworth, 260.
See Squaring of tfee elide* See Squaring of the
Byllion, 144.
Byrgius.
141, 142, 144, 222. 75.
See Biirgi.
Circlesquaring. circle.
Caesar, Julius, 90, 91. Cajori, 26, 29, 159, 237, 238, 304.
Oampano. See Campanus. Campanus, 134, 135, 136, 245, Oami^eH, X., 137. Orator, G., 70. Ctefcor, M., 5,
247.
Circulating decimals, 200, Clairaut, 269, 276, 277. Clausen, Th,, 242, ClaYius, 71, 137, 176 266. Clifford, W. K., 71, 258, 275, 281. Cloff, 198.
6, 7, 8, 10, 12, 14, 16,
192$, 31, 34, 38, 4345, 47, 50, 56, 50, 60, 67, 76, 78, 80, 81, 8385, 87, 90, 94, 98102, 105, 110113, 118, 120,
Cocker, 31, 173, 184, 190, 191, ISStai* 196, 199, 202, 203, 210, 212, 217.
Colburn,W.,218,219. Colebrook, 96.
INDEX
314Colla, 224.
Daboll, 218.
Combinations, doctrine of, 237. Commercial School in England, 179
Dase,
Data
204.
Common
logarithms, 159, 162, 165,
199.
Common
67, 88, 127. 242.
Damascius, 16fi,
of Euclid, 74, 133, 246.
Da vies,
189.
Davies, C., 277. notions, 70.
Compasses, single opening of, 128, 248, 264, 265. See Kuler and corn
Da vies, F. S., 2*JO. Da Vinci, Leonardo, De
141, 248, 249.
Benese, 178.
Decimal fractions, 150155,
Complementary 147;
117,
199, 200, 214.
See Imag
Decimal point, Decimal scale,
division,
116, multiplication, 147.
Complex numbers,
244.
inaries.
85, 152, 163,
Compound interest, 151. See Interest. Computation, modes
in
of,
Ahmes,
2124; in Greece, 27, 28, 32, 3742; in India, 95100; in Arabia, 104 106; Middle Ages, 111, 112, 114122; Modern Times, 145167, 186, 187, 206; in England, 205, 206, 241; in the United States, 241.
Computes,
112.
Coiate, v.
Conant, l 3, 4, 5. Conic sections, 63, t
78, 254, 255.
Conjoined proportion.
See Chain
Continued fractions,
150, 200.
Continuity, law of, in algebra, 237, 287 in geometry, 237, 252, 268, 287. ;
Cooley, 267. Cosine, 124, 162. Cossic art, 228. See Algebra. Cotangent, 130, 137, 163. Cotes, 208, 256.
153, 154, 194, 195. 1, 2, 3, 4, 5, 8, 11, 40,
1(55,
166, 178.
Dedekind, 72. Dee, John, 246. Defective numbers, 29,
113.
Degrees, 10, 179. De Jonquieres, 265.
De Lagny, 241, 242. De Lahire, 252, 253. De la Hire. See De
Lahire. Delboeuf , 269. Del Ferro. See Ferro. Delian Problem, 55. See Duplication of the cube.
Democritus, 45, 46.
De Morgan,
rule.
166, 187,
55, 64, 65, 68, 72, 122, 141,
151, 154, 164, 175, 178, 181, 187190, 199, 200, 205, 208, 210, 213, 238, 240, 241, 244, 246, 268, 283, 284, 286.
Demotic writing,
7.
Desargues, 237, 252, 253, 255, 257, 269. Desargues' Theorem, 253. Descartes, 233, 234, 235,236, 238, 243, 252, 254, 256, 265.
Counters, 115, 118, 185187.
Descartes' Rule of Signs, 236.
Crelle, 227, 238, 260, 261. 248, 278, 279.
Devanagarinumerals, 14, 16, 17. Dilworth, 160, 174, 191, 194, 195, 198,
Cremona,
Cridhara, 95, 102.
201, 202, 203, 217, 218, 219, 220.
Crossratio, 255, 258. Ctesibius, 80.
Dinostratus, 63. Diogenes Laertius, 28, 47, 61.
Cube, duplication of, 53, 54, 56, 226. root, 101, 106, 150, 228, 241.
Diophantine analysis, Diophantus, 26, 34r37,
ubic equations, 110, 120, 129, 224226,
104, 108, 109, 127.
Cube
227, 229, 230, 240.
Cubical numbers, 32. Cubit, 375.
Cunn,
55, 200.
Curtz, 134, 222. Cyzicenus, 63.
37, 101. 82, 87, 101, 102
Direction, concept of, in geometry, 268, 280, 289. Dirichlet, 31.
Divisio aurea, 117. Divisio ferrea, 117. Division, 37, 39, 98, 101, 105, 114, 116
INDEX 118, 140, 148, 149, 150, 151, 154, 156,
315
270, 275289;
editions of the Ele
ments, 245, 246, 266, 282, 288. Euclid and his modern rivals, 267, 2C8,
183, 189, 192, 213, 214. Dixon, E. T., 68, 268.
See Regular solids. Dodecaedron, 51 Dodgson, 267, 368, 270. Dodson, 195, 199. Dollar mark, 223. Double position, 106, 196, 198, 203. .
275289.
Eudemian Summary, Eudemus, 47, 52. Eudoxus,
4749, 61, 63, 64
46, 60, 62, 63, 64, 66.
Euler, 36, 157, 195, 200, 238, 243, 259, 265.
See False position. Dschabir ibn Aflah, 107,
Eutocius, 26. See Square root, Cube Evolution. root, Roots. Excessive numbers, 29, 111.
129, 130.
Duality, 258. Diicange, 143.
Duhamel, 77, 278. Duodecimal fractions, 3841, 119. Duodecimal scale, 2, 3, 112, 117. Dupin, 257. Duplication of the cube,
Exchange, 121, 198. Exchequer, 174, 186, 215. Exhaustion, method
53, 54, 56,
of, 59, 60, 62, 63,
66.
Exhaustion, process of, 57, 59. Exponents, 122, 152, 183, 234238.
226.
Durege, 266. Diirer, 221, 248, 249, 266.
Fallacies, 74.
Easter, computation of time of, 40,
See Computus. Egyptians, 1, 6, 7, 1926,
False position, 24, 36, 99, 106, 141, 184, 185, 196, 198, 240.
112.
27, 4446,
49, 52, 55, 76, 8890, 99, 122, 131, 170, 175.
Farrar, J., 243, 277. Farthing, derivation of word, 170. Fellowship, 121, 185.
Fenning, 191, 217. Fermat, 209, 252.
Eisentohr, 19.
Elefuga, 137.
See Euclid's
Elements of Euclid. Elements.
Ferrari, 225, 226, 228. Ferreus. See Ferro.
259, 262.
Ferro, 224, 225.
EnestrSm, 141. Engel and Staekel,
Feuerbach, 260.
Emmerich,
68, 70, 72, 267,
269272, 274.
Epicureans, 74. Equality, symbol for, 184, 195. Equations, linear, 23, 33, 37, 107, 108; quadratic, 36, 101, 102, 108, 110, 230
;
Feuerbach
circle, 259, 260.
Finseus, 147, 150, 151.
Fingerreckoning, 1, 2, 17, 27, 40, 112. Fiore, Antonio Maria. See FteMu,
cubic, 110, 120, 129, 22^227, 229, 240; simultaneous, 35; higher, 226, 227, 230,239 numerical, 240 theory of, 230233, 236, 243. Eratosthenes, 31, 56, 65.
Flamsteed, 242. Florido. S&e Floridas.
Eton, 204, 205, 209.
Floridus, 224, 225.
;
;
Euclid, 31, 35, 46, 47, 50, 52, 60, 62,
6V
64r75, 76, 77, 79, 80, 83, 85, 86, 104,
See Nine
point circle. Fibonacci. See Leonardo of Pisa. Figure of the Bride, 128.
Fisher, George, 217, 192. Fishe*, G. E., 266.
Foot, 175, 176. Fourier, 240.
134, 135, 272, 274, 276, 277, 278.
Euclid of Megara, 65, 246. Euclid's Elements, 30, 6575, 79, 81, 84, 87, 88, 91, 103, 126, 127, 133, 136
138, 206, 207, 212,
23,
245, 246, 254,
181183, 185, 207. tions.
Francais, 243.
See Decimal frac
INDEX
316
Friedlein, 8, 14, 26, 32, 3740, 64, 116,
Greeks,
117, 147. Frisius. See
91, 94, 102, 123, 126, 171, 175, 248, Greenwood, I., 217.
GemmaFrisius.
1, 4, 7, 8, 10, 20,
Greenwood,
Froebel, v. Frontinus, 91, 92.
Furlong, 177. Galileo, 155, 250.
2637, 4689,
26a
S., 217.
Gregory, D., 245. Gregory, D. F., 244. Gregory, James, 242. Groat, 178 derivation of word, 170. Gromatici, 8992. Grube, 212. Grynseus, 245. Gubarnumerals, 13, 14, 16, 17, 110. Guinea, derivation of word, 170. Gunter, 163, 179, 199. Gunther, 8, 13, 17, 117, 132, 133, 135, ;
Galley method of division, 148, 149. See Scratch method. Gallon, 172, 176, 177. Garnett, 16. Gauss, 31, 36, 74, 244, 247, 273, 274, 279,283. Geber, 107. See Dschabir ibn Aflah. Gellibrand, 163.
Geminus, 52, 84. GemmaFrisius, 203. Geometry, in Egypt, 4446
;
in
Baby
137, 159, 194, 220, 221, 247, 248, 250, 266.
Hachette, 257.
lonia, 43; in Greece, 30, 4689; in Rome, 8992; in India, 93, 122124;
Haddschadsch ibn Jusuf ibn Matar,
in Arabia, 125129; Middle Ages, 131138; in England, 135, 205,206,
Halifax, John, 122.
208, 245, 246, 259, 260, 263, 267, 281
Halliwell, 122, 133, 135, 181.
Modern Times, 230, 245289; modern synthetic, 252259 editions
Halsted, 67, 68, 70, 71, 74, 77, 246, 266,
289;
;
of Euclid, 245247, 266, 282, 288; modern geometry of triangle, 259263; nonEuclidean geometry, 266275; textbooks, 275289. Gerard of Cremona, 118, 133, 134.
126.
Hallam,
228.
268, 270275.
Hamilton, Hankel, 5,
W.
R., 55, 245.
W., 304.
26, 36, 40, 47, 50, 51, 52, 56,
59, 6062, 70, 72, 89, 90, 96, 97, 100,
Gerbert, 106, 114119, 131, 132, 147.*
103, 105107, 116, 121, 129, 132, 133, 137, 143, 230, 240, 244, 287, 288. Hardy, A. S, 243.
Gergonne, 253, 258, 260.
Harpedonaptae,
Gerhardt, 118, 167, 194. Gibbs, J. W., 245. Girard, A., 153, 230, 231, 233, 234, 247.
Harriot, 157, 231, 232, 234.
Glaisher, 163.
Golden rule, 185, 189, Rule of Three. Golden section, 63.
Gow,
193, 201.
See
Grammateus,
Gramme,
141.
171.
Grassmann,
245.
Graves, 55, 241.
Grebe point,
262.
;
Hassler, 215. Hastings, 205. '
50, 5557, 6062, 65, 67, 76, 78, 81, 85, 133, 136.
173, 174.
204, 207.
Harfin arRaschid, 126.
Hatton, 156, 195, 199, 202.
19, 20, 26, 29, 3133, 36, 44, 47,
Graap, 50. Grain of barley, 170
Harrow,
45.
of wheat, 172,
Hawkins, 190, 193. Hayward, R. B., 241. Heath, 3336. Hebrews, 44, 49, 175. Heckenberg, 144. Hegesippus, 222. Heiberg, 30, 67, 69, 75, 84. Heiberg and Menge, 70, 71, 246. Helicon, 63.
Helmholtz, 275. Hendricks, 227. Benrici and Treutlein, 280.
317
INDEX Henry L, 175, 186. Henry VII., 167. Henry VIIL, 169, 172,
Hoiiel, 278.
Hughes,
Hume,
Heppel, v, 206. Hermite, 227.
Hermotimus,
167, 168. 127.
Hunain,
Hunain ibn Ishak,
63.
Herodianic signs, Herodianus, 7. Herodotus, 27*
Heron
T., 232.
Hultsch, 65, 81.
246.
7.
of Alexandria, 7982, 90, 102,
113, 127, 250.
Heron the Elder.
See
Alexandria. Heron the Younger,
Heronie formula,
Heron
of
277.
80.
80,
127.
Hutt, 264. Hutton, 189, 243, 244, 264. Huxley, 283, 306. Hypatia, 87. Hyperbolic logarithms, 166, 167. See Natural logarithms. Hypothetical constructions, 73, 74,
123, 128,
90,
Hypsicles, 28, 31, 67, 79, 127.
134,
lamblichus, 27, 30, 33, 82, 108.
Heteromecie numbers, 29. Hexagramma mysticum, 255. Hieratic symbols,
Hieroglyphics,
Ibn Albanna, 106, 111, 150. Imaginary geometry, 272. See NonEuclidean geometry. Imaginary quantities, 227, 231233,
7.
6, 45.
Hill, J., 199, 201, 203. Hill, T., 219.
^Hlndu notation,
236,243245.
11, 1318, 145, 154,
155.
Hindu numerals,
1218, 107, 119, 121,
179, 180, 181, 187. proof, 106.
Hindu the
See Casting out
quantities. India. See Hindus. Indices.
9's.
Hindus,
Inch(uncia),40,171, 175. Incommensurables, 63, 66, 72, 266> See Irrational 278, 280, 285288.
8, 9, 11, 12, 15, 24, 82, 88,
93
103, 113, 116* 119, 122124, 128, 136,
See Exponents.
Infinite, 59, 70, 136, 250, 252, symbol for, 237.
278;
See Series.
143, 145, 147, 149, 150, 170, 215, 221, 227, 233, 238, 250. Hipparcshns, 79, 84.
Infinite series.
Hippasus, 51, 52. Hippias, 55, 56. Hippocrates of Chios, 56, 57, 59, 60,
Interest, 100, 121, 151, 167, 168> 198.
Infinitesimal, 136. Insurance, 200.
Hirst, 279, 284, 286. Hodder, 191, 217.
Inversion, 99. Involution of paints, #T, 258, 253. Ionic School, 4740. * Irrational qn^ titles, *^ *t, T , 102, liSt 108^ 287, JSfee la*
Hoefer, 133. Hoernle, 94.
Mdoms, 88.
131,657^
of Carthage, 92y Ml. Isoperimetrical figures, 79>, 136. Italian method of division, 21&
Mdorus
Hoffmann, 50, 128. Hoffmann's Zeitschrift, 280. Holywood, 122. Homologieal figures, 253. Homology, law of, 288.
Jakobi, C. F, A., 260, 26L
Joachim, 251 See Btoteia. John of Palermo, 120.
Horace, 40.
Homer, 226, 240. Homer's method, Homer.
John of Seville, 240,
241.
See
Johnson,
3.
Jones, 283,
118, 119, 15CL
,
INDEX
318
Leudesdorf 248. Leyburn, 203.
Jones, W., 200. Jonquieres, de, 265.
,
Jordanus Nemorarius, 134, 135, Joseph Sapiens, 117.
142.
Limits, 77, 78, 250, 278, 287.
Lmdemann,
Joseph's Play, 222. Josephus, 222.
243.
Linkages, 263, 264. Lionardo da Vinci.
See Vinci.
Lippich, 265.
Kambly,
280.
Littre', E., 144.
Kant, 277, Kastner, 127, 181, 207, 279.
Lobatchewsky,
Kelly, 168, 169, 170, 174, 196.
Local value, principle
Kempe,
86, 127, 266, 272275,
279.
264.
Kepler, 136, 155, 166, 167, 237, 247, 249, 252, 269.
Kersey, 153, 194, 196, 199, 201, 203, 220,221. Kettensatz, 196. See Chainrule. Kingsley, 87. Kircher, 247. Klein, 70, 74, 249, 264, 265. Koppe, K., 280. Kranse, 247.
of, 8, 9, 10, 11,
12, 15, 42, 95, 105, 118. Locke, 144.
Logarithmic series, 167. Logarithmic tables, 160166. Logarithms, 155167, 187, 199,^229. Lorenz, 269. Loria, 25, 47, 48, 50, 56, 59, 65, 67, 76, 277280, 283.
Lowe,
189.
Liibsen, 279. Luca Paciuolo.
Lucas
See Pacioli. See Pacioli.
Kronecker, 227. Kruse,280.
di Burgo. Lucian, 31.
Lacroix, S. F., 268, 276. Laertins, Diogenes, 28, 47, 61.
Ludolph van Ceulen, 242. Ludolph's number, 242. Ludus Joseph, 222. Lnne, 57. Lyte, 152.
Lagny, de, 241, 242. Lagrange, 36, 214, 240, 270, 283. Lahire, de 252, 253. Lambert, 242, 248, 269, 274. Lange, J., 259. Langley, E. M., 213, 241. Langley and Phillips, 288. t
Macdonald, 156, Machin, 242.
259, 262, 288.
Mackay,
Mackay
160, 194, 206.
circles, 263.
Laplace, 155, 269.
Maclaurin, 208, 244, 256.
La Roche,
Magic squares,
142, 144. Larousse, 276. Lefere, 142.
Mahavira, 95. Malcolm, 200.
Legendre,
Malfatti, 226.
31, 243, 269, 270, 271, 272, 276, 277, 278, 283, 286. Leibniz, 195, 254, 262.
Lemma of Menelaus,
83.
Martin, B., 194, 199. Martin, H., 218. Mascheroni, 264.
Lemoine circles, 262. Lemoine point, 262. Leodamas, 63.
Mathematical recreations, 24,
Leon, 63, 65.
135, 139, 141, 150. 196, 222.
Marcellus, 75, 78. Marie, 80, 133, 199, 252, 264
Marsh, 200.
Lemoine, E., 259, 262.
Leonardo da Vinci, 141, 248, 249. Leonardo of Pisa, 24, 76, 119121,
221.
219223. ,
134,
Matthiessen, 23, 227, 234.
Mandith,
137.
MarweU,
77.
319
INDEX McLellan and Dewey,
212.
Napier, J., 149, 153, 155167, 194, 206, 229,251.
Mean
proportionals, 56. Measurer by the ell, 176. Medians of a triangle, 134.
Napoleon I., 69, 257, 264. Nasir Eddin, 127, 128, 130, 131, 270. Natural logarithms, 159, 160, 164, 166. Naucrates, 65. Negative quantities, 35, 88, 101, 102,
Meister, 247. Mellis, 188.
Mensechmus, 63. Menelaus, 82, 85,
129, 133;
lemma
of,
Menge. See Heiberg and Menge. Mental arithmetic, 204, 219. 204, 206,
See Teaching of Mathe
Metric System, 152, 171, 178, 179, 215.
226, 237240, 243, 255, 273.
Nicomachus,
for, 141, 184, 228.
See
See Tartaglia.
26, 3133, 41, 91, 92.
Niedermiiller, 214.
Ninepoint circle, 259, 260.
Subtraction.
Minutes, 10. Mobius, 247, 258. Moivre, de, 208. Moneyer's pound, 168. pound.
Nixon, 285, 286, 288. NonEuclidean geometry, 266275. See
Tower
Norfolk, 180, 181. Norton, K, 152. Notation, of numbers (see Numerals, Local value) of fractions, 21, 26, 98, 106, 181; of decimal fractions, 152154 in arithmetic and algebra, 23, ;
See
De Morgan.
Moschopulus, 221.
;
Motion of translation,
70, 135, 157; of reversion, 70; circular and recti linear, 263, 264.
Muhammed
68, 288.
55, 77, 167, 170, 194, 208, 209,
Nicolo Fontana.
Meziriac, 220, 222. Million, 143, 144.
Monge, 257. Morgan, de.
30, 31, 37, 72, 108, 109. circles, 263.
Newcomb, Newton,
matic's.
Minus, symbol
See Jor
Nesselmann,
Neuberg
207.
Methods.
142.
Neocleides, 63. Neper. See Napier.
Mercator, N., 167.
Merchant Taylors' School,
107, 227, 230233, 243, 244, 258.
Nemorarius, 134, 135, danus Nemorarius.
83.
ibn
Musa Alchwarizmi,
104109, 118, 128, 129, 133, 206.
30, 35, 36, 98, 101, 108111, 122, 184, 194, 195, 228, 229, 234, 235, 237, 239, Number, concept of, 29, 35, 232, 233.
See Negative quantities, ImaginaIncommeasTiIrrationals,
ries,
rabies.
Miiller, F., 107. Miiller, H., 268, 280. Miiller, John, 250.
Numbers, amfcable, 29; See Regiomon
tanus. Multiplication, 26, 9698, 101, 105, 110, 115, 140, 145147, 156, 189, 194, 195, 213, 214, 234; of fractions, 21, 22, 181, 182.
Multiplication table, 32, 39, 40, 115, 117, 147, 148, 181. 173.
cubical, 3S; 113; excessive, 29, 111 ; heteromeeic, 29 ; prime, 30T 31, 74; perfect, 29, 111, 113; polygonal, 31,32; triangular, 29, defective, 29,
Numbersystems, 118, Numerals, 6, 7, 8, 1216,
32, 38, 107,
119, 121, 180.
Numeration, 15, S, 1218, 142145.
Murray,
Musa ibn SchaMr,
128.
Mystic hexagon, 255.
Observation in mathematics, 74, 78, 151, 155, 283, 288.
Octonary numeration,
Naperian logarithms, 159. Kapler, M., 154.
3.
OEnopides, 46. Oinopides. See OEnopides.
77,
INDEX
320 Ohm,
Peuerbaeh. See Purbach. Peurbach. See Purbach.
M., 244.
Oldenburg, 237. Omar Alchaijami, 110. Oresme, 123, 234. Otho, V., 251. *
Peyrard, 71. Philippus of Mende, 63. Philolaus, 29, 53. Phoenicians, 208.
Ottaiano, 87.
Picton, 180. Pike, 214, 216218.
Oughtred, 149, 153, 154, 187, 194, 195, 203,234.
Ounce
Pitiscus, 203, 251.
(uncia), 40, 112, 171, 172, 173.
Oxford, university
of, 137, 138.
Planudes, *
IT,
44, 49, 73, 76, 123, 128, 145, 241243.
See Squaring of the circle. Pacciuolus. See Pacioli.
16, 95, 213.
Plato, 26, 29, 46, 50, 53, 54, 56, 6063, 65, 114, 208.
Pacioli, 105, 139, 141143, 145, 146148, 181, 182, 188, 196, 208, 214, 222, 232,
Plato of Tivoli, 118, 124, 130, 134. Plato Tiburtinus. See Plato of Tivoli. Platonic figures, 65, 73, 78. Platonic school, 53, 6063.
238,245. Paciuolo. See Pacioli.
Playfair, 270, 271, 282. Plus, symbol for, 141, 184, 228.
Padmanabha,
Palatine anthology, Palgrave, 186, 256.
33, 34.
Palms, 175. Paolis, E. de, 279. Paper folding, 265. Pappus, 65, 79, 82, 8689, 248, 255. Parallel lines, 43, 52, 85, 86, 127, 252, 266275, 280, 282. Parallelpostulate, 70, 71, 85, 86, 127, 266275, 279, 282, 283, 289; stated, 71.
Parley, Peter, 218. Pascal, 238, 252255. Pascal's theorem, 255, 257. Pathway of Knowledge, 173, 174, 188, 189.
Pedagogics.
See Teaching.
Peirce, B., 245. Peirce, C. 8., 67, 70. Peletier, 230, 266.
Pentagramstar, 136. Pepys, 192. Perch, 178. Perfect numbers, 29, 111, 113. Perry,
Plutarch, 47. Poinsot, 247. Polars, 257, 258.
Polygonal numbers, Poncelet,
248,
31, 32.
252,
257,
253,
258,
260.
Pons asinorum, 137. Porisms, 74. Porphyrius, 82. Position, value.
principle
of.
See Local
Postulates, 46, 54, 61, 67, 69, 70, 71, 73, 89, 134, 266275, 279, 281.
Pothenot, 251. Pothenot's problem, 251. Pott, 4.
205, 210. Peacock, 2, 121, 143, 145150, 153, 181, 182, 186, 187, 193, 198, 220, 244. Peaucellier, 263, 264.
Peacham,
Perier,
See
Addition.
95.
Madam, J.,
254.
291.
Pestalozzi, v, 197, 203, 211, 212, 219. Petrus, 136.
40, 168175, 178, 185, 218. Practice, 22, 189, 196.
Pound,
Primes, 30,
"
31, 74.
Problems for Quickening the Mind," 113, 131, 220.
Proclus, 47, 61, 63, 70, 74, 85,
88,
247.
Progressions, arithmetical,
9, 23, 24,
31, 100, 158, 160, 181, 185, 237;
metrical,
geo
9, 23, 24, 100, 158, 160, 181,
237.
Projection, 258. Proportion, 30, 31, 47, 53, 57, 65, 67, 72, 73, 110, 196, 197, 202, 234, 277f 282, 285, 286. See Rule of Three.
Ptolemaeus.
See Ptolemy.
INDEX Ptolemy,
28, 82, 84, 104, 123, 124, 126, 129, 130, 270. Ptolemy I., 64.
Public schools in England, 204.
Purbach, 140, 148, 149, 213, 250. Pythagoras, 46, 4853, 123, 136; theo
rem of, 45, 4951, 56, 66,
123, 128, 137.
school, 4953.
Pythagorean
Pythagoreans,
14, 27, 29, 30, 4953, 66,
136,250.
321
Rigidity of figures, 280. Rigidity postulate, 71. Roberval, 209. Robinson, 216, 267. Roche. See La Roche.
Rodet, 94.
Rods, Napier's, 156, 157. Rohault, 65.
Roman
notation, 4, 8, 11, 121, 180. 1, 4, 8, 20, 3742, 82, 8892
Romans,
116, 122, 131, 135, 168, 175. A., 242, 251.
Romanns, Quadratic equations. See Equations, Quadratrix, 55. Quadrature of the circle. See Squar ing of circle. Quadratures, 167, 237. Quadrivium. See Quadrnvium.
Quadruvium,
92, 111, 114.
Quick, v, 212.
Roods, 178.
Root
(see
Square root, Cube
root),
36, 72, 76, 101, 102, 106, 108, 110, 119, 150, 151, 200, 227233, 235, 236, 241.
Rosenkranz, 208. Rouche and C. de Comberousse, 278, 287.
Quinary
scale, 1, 3, 4. Quotient, 37, 115, 183.
Row,
Radian, 251. Radical sign, 228, 229, 235,
Rugby,
S., 265.
Rudolff, 141, 143, 151, 228, 229, 232. Ruffini, 226.
Raleigh,
W.
f
236.
231.
247, 275.
Ratio, 32, 51, 63, 69, 76, 87, 167, 188, 194, 195, 234.
Raumer,
184.
See False
position. Rule of four quantities, 129. Rule of six quantities, 83, 129.
Ralphson, 55.
Ramus,
204, 207.
Rule of Falsehode,
Rule of Three,
100, 106, 121, 185, 193See Proportion. 196, 198, 201, 202.
Ruler and compasses,
286*
Reciprocal polars, 258. Recorde, 118, 147, 149, 183188, 194, 196, 198, 206, 208, 228, 234, 281.
54, 135, 248, 249,
264.
Ruler, graduated, 134. Rutherford, 242.
Reddall, 204, 207.
Reductio ad absurdum, 58, 60, 62* Rees, A., 166. Rees, K. F., 196, 197. Reesischer Satz, 196. See Chainrule.
Regiomontanus,
108, 136, 140, 181, 208, 229, 247, 250, 251.
Regular
solids, 52, 62, 65, 67, 73, 78, 128, 136, 249. Reiff , 238.
Saccheri, 267, 269, 271, 274, Sacroboseo, 122.
Sadowski, 148, 213, 215. Salvianus JmHamas, 41. Sandcounter, 29. Sannia and E. D'Oridio, 279, 28? . Sanscrit letters, 16.
Rhsefcicus, 187, 251.
Savile, 281, 282.
Rheticos, See Rhaeticras. R&etorieal algebras, 108, 120. Rhind papyrus, 19. See Ahmes. Richards, E, L., 74, 268.
Schilke,79. Schlegel, 280.
Riemann, Rkse, Adam,
Schmid, K.A^ 143, 28d. Schmidt, F., 273.
31, 275, 283. 31, 142, 192, 196,
2ia
2f&
Schlussrechnnng, 196*197. ysis in arithmetic.
INDEX
322 Schotten, H., 230. Schreiber, H., 141. Schroter, 265. Schubert, H., 249. Schulze, 166.
Sphere, 52, 62, 63, 78, 79, 249. Spherical geometry, 73, 83, 125, 156, 247. See Sphere.
Square root,
Schwatt, 266. Scratch method of division, 148, 149, 192, 215; of multiplication, 105.
Scratch method of square root, 150. Screw, 55. Secant, 130, 137, 251. Seconds, 10. Seueca, 65. Serenus, 8284. Series, 32, 110, 120, 167, 229, 238, 239, 242, 243. See Progression. Serret, 227. Servois, 248, 257.
Sexagesimal fractions,
10, 20, 106, 119,
120, 151.
Sexagesimal scale,
6, 9, 10, 28, 43, 79,
84, 85, 124, 163.
Sextus Julius Africanus, Shanks, 145, 242. Sharp, A., 241, 242.
28, 76, 101, 106, 110, 150, 151, 154, 185. Squaring of the circle, 54, 55, 57, 58. 200, 226, 249, 271, 282. See w.
See Engel and Stackel.
Stackel.
Starpolysedra, 247, 250.
Starpolygons, 136, 247. Staudt, von, 258, 259, 265. Steiner, 248, 258, 264.
Stephen, L., 133. Sterling, 172 ; origin of word, 170. Stevin, 122, 149, 151153, 157, 234, 23l<. Stewart, M., 256. Stifel, 141, 143, 149, 157, 158, 232, 233, 234, 238.
208 226
Stobseus, 65. Stone, 201, 240, 262, 267, 269.
Sturm, 253. St. Vincent, 167.
86.
Subtraction, 26, 37, 38, 96, 101, 105, 110, 115, 145.
Sharpless, 204, 207. Shelley, 173, 187, 199. Shillings, 168, 169, 170, 185, 218. Sieve of Eratosthenes, 31. Simplicius, 58.
Simpson, T., 243, 244, 267, 284. Simson, E., 67, 69, 71, 74, 246, 256, 282,285.
Surds, 150. Suryasiddhanta, 12. Surveying, 89, 91, 136, 178. Suter, 128130, 133, 137. Sylvester, 22, 264, 283, 285.
Symbolic algebras,
Symmedian
109.
point, 259, 262.
Syncopated algebras, 108. Synthesis, 62.
Sine, 85, 124, 129, 130, 159, 162, 163, 251 ; origin of word, 124, 130.
Singhalesian signs, 12. Single position, 196. See False posi tion.
Tabit ibn Kurrah, 127, 129. Tables, logarithmic, 160166; multi plication, 32, 39, 40, 115, 117, 147, 148, 181 ; trigonometric, 79, 84, 124, 125, 130, 159, 250, 251. Tacquet, 66, 188.
Slack, Mrs., 192, 217. Sluggard's rule, 147. Snellius, 251.
Socrates, 60, 65.
Tagert, 70.
Sohncke, 83.
Tangent in trigonometry,
Solids, regular. See Regular Solon, 7, 28. Sophists, 26, 5360, 276. Sosigenes, 91. Speidell, E., 165. Speidell,
J.,
164, 165.
Spencer, v. Spenser, 172.
solids;.
130, 137.
250,251.
Tannery,
47, 56, 81, 128.
Tartaglia, 139, 145, 150, 188, 193, 196, 198, 208, 220, 222, 224, 225, 228, 248.
Taylor, B., 208. Taylor, C., 252, 257, 264. Taylor, H. M., 68, 288.
Taylor
circles, 263.
INDEX Teaching, hints on methods
of,
v,
17, 25, 41, 74, 78, 114, 138, 151, 155,
183, 185, 189, 197, 198, 211, 212, 233, 236, 238, 239, 277, 281, 288. Telescope, 155, 232.
Thales, 4649, 52.
Van
323 Ceulen, 242.
Varignon, 267. Varro, 91. Vasiliev, 272.
Venturi, 81. Veronese, G., 279.
Theaetetus, 63, 64, 66.
Versed
Theodoras, 60. Theodosius, 133. Theon of Alexandria,
Vesa, 242. 28, 69, 79, 82,
87, 91.
Theon of Smyrna, 33, 82. Theory of numbers, 37, 51, Theudius,
Victorius, 40. Vieta, 79, 109, 187, 226, 228232, 233 234, 238, 240, 242, 250, 251. Vigarie', E., 259.
82, 112.
63, 65.
Thompson,
sine, 124.
Vigesimal scale,
2, 3, 4.
Villefranche, 142. Vinci, Leonardo da, 141, 248, 249.
T. P., 279.
Thymaridas, 33, 108. Timaens of Locri, 60.
Vlacq, 163, 167. Von Staudt, 258, 259, 265.
Timbs, 204, 205, 207, 210. Todhunter, 62, 68, 69. See Abu'l Wafa.
Tonstall, 16, 143, 149, 180183, 185, 187, 188, 203, 206, 208.
Wagner,
Torporley, 251. Tower pound, 168, 171, 172, 174. Transversals, 83, 252, 255, 258.
Walker, F. A., 169. Walker, J. J., 157. Wallace line and point,
Treutlein, 13.
Wallis, 123, 128, 149, 153, 164, 200,
Triangular numbers, 29. Trigonometry, 79, 81, 84,
135,226. 114.
Wantzel, 227.
Ward,
200, 209.
Warner, W., 232. Webster, W., 210. Weights and measures, 166179
169, 170, 172, 174; deri vation of word, 172.
;
in
Weissenborn,
Tucker, 262.
Wells, 205. Wessel, C, 243.
Tylor,
263.
3, 4, 17, 204.
Tyndale, 183.
16, 121, 140, 142, 144, 147, 148,
181, 191, 194, 195, 204, 207, 212, 215.
University of Oxford, 137, 138. University of Paris, 137. University of Prague, 137.
Usury, 168,
Vacquant, Ch., 278, Valentin, 87.
76, 106, 117, 255.
.
Whewell, 281. Whiston, 66, 239. Whitehead, 246. Whittey, 260. Whitney, 12.
Widmann, Unger,
\ 9, 140, 152,
United States, 215217.
Troy pound,
Tucker circle, 262, Tycho Brahe, 250.
259.
208, 209, 236, 237, 269, 274. 85, 123125,
129131, 136, 139, 155, 156, 159, 165, 230, 243, 250, 251, 270, Trisection of an angle, 34, 55, 134,
Trivium, 92,
U., 140.
140, 141, 196.
Wiener, H., 265. Wilder, 194.
^
Wildermuth,
143, 153, 154, 189, 195,
222.
William the Conqueror,
168, 171.
Williamson, 246. Wilson, J. M., 268, 283,
Winchester bushel, 176. See Bushel Winchester (school), 204, 206. Windmill, 137.
INDEX
324 Wingate,
153, 173, 187, 194, 195, 196, 199, 201, 202, 203, 220, 221. Wipper, 50, 128.
Yancos, 145. Yard, 175, 176.
Young,
6.
194, 266. Wolfram, 166.
Wolf,
Wopcke, 14, 15. Wordsworth, C., Worpitzky, 280.
244.
Wright, 153, 200, 206.
Zambertus, 245. Zeno, 58, 59. Zenodorus, 79, 136, 247. Zero, 10, 11, 12, 14, 15, 42, 95, 99, 115 118, 119, 143, 153.
Ziwet, 280.
Xenocrates, 61.
Zschokke, 215.
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1957
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